The Bloch-Kato conjecture and a theorem of Suslin

THE BLOCH-KATO CONJECTURE AND A THEOREM OF
SUSLIN-VOEVODSKY
THOMAS GEISSER AND MARC LEVINE
Abstract. We give a new proof of the theorem of Suslin-Voevodsky which
shows that the Bloch-Kato conjecture implies a portion of the BeilinsonLichtenbaum conjectures. Our proof does not rely on resolution of singularities,
and thereby extends the Suslin-Voevodsky theorem to positive characteristic.
1. Introduction
The purpose of this paper is to give an alternate proof of the main result of
[25] along the lines of [13]. Our proof does not rely on resolution of singularities,
hence extends the results of [25] to varieties over fields of arbitrary characteristic.
The new ingredient which enables us to apply the techniques of [13] to motivic
cohomology is the surjectivity result of [8, Corollary 4.4].
Let F be a field, q ≥ 0 an integer, and m > 1 an integer prime to the characteristic of F . We have the Milnor K-group KnM (F ), and the Galois symbol
(1.1)
q
ϑq,F : KqM (F )/m → Hét
(F, µ⊗q
m ).
The Bloch-Kato conjecture [3] asserts that the map ϑq,F is an isomorphism for all
F , q and m. For q = 1, this follows from the definition of ϑ1,F via the Kummer
sequence, and is the theorem of Merkurjev-Suslin [18] for q = 2. For m a power of
2, the Bloch-Kato conjecture for arbitrary q and F has been proven by Voevodsky
[28]
Let X be a localization of a smooth scheme of finite type over k; we call such
a scheme essentially smooth over k. We use Bloch’s higher Chow groups as our
definition of motivic cohomology H p (X, Z(q)) for essentially smooth k-schemes X.
We sheafify Bloch’s cycle complexes to define the weight q motivic complex ΓX (q),
as a complex of Zariski sheaves on X.
Let m be prime to the characteristic of k, let X be essentially smooth over k,
and let : Xét → XZar be the change of topology map. We construct a natural
cycle class map (in the derived category of Zariski sheaves)
(1.2)
cla : ΓX (a) ⊗L Z/m → τ≤a R∗ (µ⊗a
m ).
Our main result is
Theorem 1.1. Let k be a field, and let m be an integer prime to the characteristic
q
of k. Suppose that the maps ϑq,F : KqM (F )/m → Hét
(F, µ⊗q
m ) are surjective for
1991 Mathematics Subject Classification. Primary 19E20; Secondary 19D45, 19E08, 14C25.
Key words and phrases. Motivic cohomology, étale cohomology, Milnor K-theory, higher Chow
groups.
The second author acknowledges support by the Deutsche Forschungsgemeinschaft and the
NSF.
1
2
THOMAS GEISSER AND MARC LEVINE
all finitely generated field extensions F of k. Then the cycle class map (1.2) is an
isomorphism for all essentially smooth X over k and all a with 0 ≤ a ≤ q.
As one consequence, we have
Corollary 1.2. Let k be a field, and let m be an integer prime to the characteristic
q
of k. Suppose that the maps ϑq,F : KqM (F )/m → Hét
(F, µ⊗q
m ) are surjective for all
finitely generated field extensions F of k. Let X be essentially smooth over k. Then
the cycle class map (1.2) induces an isomorphism
p
clq : H p (X, Z/m(q)) → Hét
(X, µ⊗q
m )
for all p ≤ q, and an injection for p = q + 1.
The conclusion of Theorem 1.1 is a part of the Beilinson-Lichtenbaum conjectures
(see e.g. [16]).
Using Voevodsky’s verification of the Bloch-Kato conjecture for m a power of
2 (loc. cit.), we may apply Theorem 1.1 to extend the consequences of [28] to
characteristic p > 0:
Corollary 1.3. For m = 2ν , the cycle class map (1.2) is an isomorphism for all
X essentially smooth over k.
Similarly, the Merkurjev-Suslin theorem (loc. cit.) gives as in [25]:
Corollary 1.4. Let k be a field, and let m be prime to char k. Then the cycle class
map (1.2) is an isomorphism for all a, 0 ≤ a ≤ 2, and all X essentially smooth
over k.
We also consider the étale sheafification ΓX (q)ét of ΓX (q). The results of SuslinVoevodsky [24] on the Suslin homology of varieties over an algebraically closed field
give us the following unconditional result:
Theorem 1.5. Let k be a field, and let m be an integer prime to char k. Let X be
essentially smooth over k. Then the étale cycle class map
clqét : ΓX (q)ét ⊗L Z/m → µ⊗q
m
is an isomorphism in D− (Sh(Xét )) for all q ≥ 0.
With the aid of this result, one can reformulate Theorem 1.1 entirely in terms
of ΓX (q) and ΓX (q)ét , giving the equivalent result which avoid the use of the cycle
class map.
Theorem 1.6. Let k be a field, and let m be an integer prime to char k. Supq
pose that the maps ϑq,F : KqM (F )/m → Hét
(F, µ⊗q
m ) are surjective for all finitely
generated field extensions F of k. Then the natural map
ιX : ΓX (a) ⊗L Z/m → τ≤a R∗ ΓX (a)ét
is an isomorphism for 0 ≤ a ≤ q and for all X essentially smooth over k.
An outline of the paper is as follows: In §2 we recall the definition of the primary
object of study, Bloch’s cycle complex, and describe its fundamental properties.
We also extend the definition of the cycle complex to the relative situation, and to
normal crossing schemes. In §3, we give a definition of the cycle class map from
motivic cohomology to étale cohomology, and we show in §4 that this defines a
natural transformation of cohomology theories, compatible with Gysin sequences
THE BLOCH-KATO CONJECTURE AND A THEOREM OF SUSLIN-VOEVODSKY
3
and products. These two sections are fairly technical, and could be skipped on
the first reading. The main argument for the proof of Theorem 1.1 occurs in §5§7. In §5, we introduce the semi-local n-cube, the relative motivic cohomology of
which provides the dimension shifting which enables us to move from H q (Z/m(q))
to H p (Z/m(q)) for p ≤ q. The heart of this dimension shifting argument is given in
§6 and §7. We prove Theorem 1.5 and show that Theorem 1.1 and Theorem 1.6 are
equivalent at the end of §4. We have included an appendix on products for Bloch’s
cycle complexes, which fills a gap in the construction of products given in [1].
We are indebted to B. Kahn for pointing out how to use Lemma 7.3 to remove
some unnecessary hypotheses from an earlier version of Theorem 1.1. We would
also like to thank the referee and the editor for making a number of useful comments
and suggestions.
2. Prelimiaries
We set up some notations and conventions, and recall some of the notations and
results from [8] for the reader’s convenience. We fix a field k.
2.1. Conventions. We will have occasion to work with both complexes for which
the differential has degree −1 (homological complexes) and those for which the
differential has degree +1 (cohomological complexes). Rather than work with two
equivalent derived categories, we will always work in the derived category formed
from cohomological complexes. If A∗ is a homological complex, the associated
cohomological object will be the cohomological complex A∗ with An := A−n , and
with dn := d−n . Similarly, we will give the category of homological complexes the
translation structure which is compatible with that of the derived category, i.e., if
A[1]
(A∗ , dA
) is the complex with
∗ ) is a homological complex, then (A[1], d
A[1]n := An−1 , dA[1]
:= −dA
n
n−1 .
We will sometimes abuse notation by saying that a sequence of complexes
i
j
S := [A −
→B−
→ C]
gives a distinguished triangle if the image of S in the derived category has an
extension to a distinguished triangle in some standard way. For instance, if S is
degree-wise exact, with i degree-wise injective and j degree-wise surjective, then
the canonical map cone(i) → C is a quasi-isomorphism, hence S defines a canonical
distinguished triangle.
We let [n] denote the ordered set {0, . . . , n}, with the standard order, and let ∆
denote the category with objects the sets [n], n = 0, 1, . . . , and with morphisms the
order-preserving maps. For a simplicial abelian group A : ∆op → Ab, we view the
associated chain complex as a cohomological complex A∗ in negative degrees, i.e.,
An = A([−n]), with differential the usual alternating sum. Similarly, for a simplicial
object A : ∆op → C(Ab), we have the double complex A∗∗ , with Aa,b = Aa ([−b]).
2.2. Cycle complexes. Let ∆n be the affine space
n
∆n := Spec k[t0 , . . . , tn ]/
ti − 1.
i=0
A subscheme of ∆m defined by equations of the form ti1 = . . . = tir = 0 is called
a face of ∆n (we consider ∆n a face of ∆n as well). The vertices vin of ∆n are the
faces tj = 0, j = i. For a map g : [n] → [m] in ∆, we have the map of schemes
4
THOMAS GEISSER AND MARC LEVINE
g : ∆n → ∆m defined by taking the affine-linear extension of the map of vertices
m
; via these maps, sending n to ∆n extends to form a cosimplicial scheme
vin → vg(i)
∆∗ .
Let X be a finite type k-scheme, locally equi-dimensional over k. We let z q (X)
denote the group of codimension q algebraic cycles on X. Bloch [1] has defined
the cycle complex z q (X, ∗), with z q (X, p) the subgroup of the group z q (X × ∆n )
generated by codimension q subvarieties W of X × ∆p such that, for each face F
of ∆p , W ∩ X × F has codimension ≥ q on X × F . For each order-preserving map
g : [n] → [m], and each W ∈ z q (X, m), the pull-back (id × g)∗ (W ) via the induced
map id × g : X × ∆n → X × ∆m is defined and is in z q (X, n). Thus, sending
m to z q (X, m) gives rise to a simplicial abelian group, and z q (X, ∗) is defined as
the associated chain complex. The higher Chow groups of X, CHq (X, p), are then
defined as the homology
CHq (X, p) := Hp (z q (X, ∗)).
In fact, the definition given in [1] requires X to be quasi-projective, but this is
not necessary. More generally, if X is the localization of a locally equi-dimensional
finite type k-scheme Y , then define
z q (X, ∗) =
lim
→
z q (U, ∗),
X⊂U ⊂Y
where the limit is over open subschemes U of Y . It is easy to see that z q (X, ∗)
is independent of the choice of Y . We set as above CHq (X, p) := Hp (z q (X, ∗)).
In particular, all the notions described above make sense for a k-scheme X of the
form Spec(OX,v ), where OX,v is the local ring of a smooth quasi-projective variety
X at a finite set v of closed subvarieties.
2.3. Pull-back. As we will see below, the higher Chow groups have the formal
properties of Borel-Moore homology for schemes essentially of finite type over k,
and, for essentially smooth k-schemes, the formal properties of cohomology, including contravariant functoriality. The complexes z q (X, ∗) are not, however, contravariantly functorial for arbitrary maps f : Y → X, even if X and Y are smooth,
due to the fact that the pull-back of an arbitrary cycle on X × ∆m is not in general defined. As a first step in the construction of a pull-back map f ∗ , we need to
consider subcomplexes of z q (X, ∗) constructed by using cycles having good intersections with a given finite collection of closed subsets of X.
Let X be a k-scheme, essentially of finite type and locally equi-dimensional over
k, and let S be a finite set of closed subsets of X, including X as an element. We
have the subcomplex z q (X, ∗)S of z q (X, ∗) with z q (X, p)S being the subgroup of
z q (X, p) generated by subvarieties W which intersect S × F properly on X × ∆p ,
for all S in S, and all faces F of ∆p .
Let f : Y → X be a morphism of locally equi-dimensional k-schemes, essentially
of finite type, and let
(2.1)
Si (f ) = {x ∈ X| dimk(x) f −1 (x) ≥ i},
S(f ) = {Si (f )|i = −1, 0, 1, . . . }.
Then each Si (f ) is a closed subset of Y . Suppose that X is regular and that
S(f ) ⊂ S. Then (f ×id)∗ (W ) is well-defined and in z q (Y, p) for each W ∈ z q (X, p)S ,
THE BLOCH-KATO CONJECTURE AND A THEOREM OF SUSLIN-VOEVODSKY
5
giving the map of complexes
(2.2)
fS∗ : z q (X, ∗)S → z q (Y, ∗).
If X is smooth over k and affine, then the proof of [14, Chap. II, Theorem 3.5.14],
shows that the inclusion
(2.3)
iS : z q (X, ∗)S → z q (X, ∗)
is a quasi-isomorphism. Setting f ∗ = fS∗ ◦ i−1
S defines a pull-back map in D(Ab),
(2.4)
f ∗ : z q (X, ∗) → z q (Y, ∗),
and the resulting map in cohomology f ∗ : CHq (X, ∗) → CHq (Y, ∗).
If, for example, f is an inclusion iY : Y → X and Y is in S, the map (2.2) is
defined. Let S(Y ) be the subset of S consisting of those S contained in Y , and
write z q (Y, ∗)S for z q (Y, ∗)S(Y ) . One sees directly that i∗Y (z q (X, p)S ) ⊂ z q (Y, p)S ,
giving the map of complexes i∗Y : z q (X, ∗)S → z q (Y, ∗)S .
2.4. External products. We refer the reader to the Appendix for a description
of natural external products
z q (X, ∗) ⊗L z q (Y, ∗) → z q+q (X ×k Y, ∗)
in D− (Ab). We will show in §2.10 how the external products give rise to natural
cup products for X essentially smooth over k.
2.5. Motivic cohomology. The higher Chow groups have been incorporated into
a general theory of motivic cohomology via the categorical constructions of Voevodsky [27] and the second author [14]. In case resolution of singularities holds for
finite type k-schemes (e.g., k has characteristic zero), the motivic cohomology of
[27] and [14] are canonically isomorphic (see [14, Chap. IV, Theorem 2.5.5]); for
smooth X which is the localization of a quasi-projective k-scheme both theories
agree with the higher Chow groups via a canonical isomorphism
(2.5)
CHq (X, p) ∼
= H 2q−p (X, Z(q)).
Even without assuming resolution of singularities, the isomorphism (2.5) holds for
the motivic cohomology of [14] (see [14, Chap. II, Theorem 3.6.6]).
We reindex Bloch’s cycle complex to reflect the isomorphism (2.5), defining the
cohomological cycle complex Z q (X, ∗), for X essentially smooth over k, by
Z q (X, p) := z q (X, 2q − p),
and take the cohomology H p (Z q (X, ∗)) as the definition of the motivic cohomology H p (X, Z(q)). We define the subcomplex Z q (X, ∗)S of Z q (X, ∗) by the similar
reindexing of z q (X, ∗)S .
Let m be an integer, and X essentially smooth over k. The mod m motivic
cohomology, H p (X, Z/m(q)), is defined by
H p (X, Z/m(q)) := H p (Z q (X, ∗) ⊗ Z/m).
The mod m motivic cohomology has a natural definition via the motivic categories
of [27] and [14] as well, which agrees with the definition given here, subject to the
restrictions described above.
6
THOMAS GEISSER AND MARC LEVINE
2.6. Localization. Let X be a locally equi-dimensional k-scheme, essentially of
finite type over k, and let W be a closed subset of X, with inclusion i : W → X,
and open complement j : U → X. We set
q
zW
(X, ∗)S := cone(j ∗ : z q (X, ∗)S → z q (U, ∗)S )[−1].
In case X is smooth over k, we set
q
ZW
(X, ∗) := cone(j ∗ : Z q (X, ∗)S → Z q (U, ∗)S )[−1],
(2.6)
and define the motivic cohomology with support as
p
q
HW
(X, Z(q)) := H p (ZW
(X, ∗)).
Suppose now that W has pure codimension d on X, giving us the push-forward
i∗ : z q−d (W, ∗) → z q (X, ∗). Since j ∗ ◦ i∗ = 0, we have the canonical map
q
q−d
iW
(W, ∗) → zW
(X, ∗).
∗ :z
(2.7)
For X quasi-projective, it is shown in [2] that iW
∗ is a quasi-isomorphism. In fact,
the argument of [2] never uses the fact that X is quasi-projective, and shows that
iW
∗ is a quasi-isomorphism for X of finite type over k; a limit argument shows that
iW
∗ is a quasi-isomorphism for X essentially of finite type over k as well. For details
on this extension of the results of [2], we refer the reader to [15].
Combining iW
∗ with the standard cone sequence gives us the distinguished triangle
j∗
i
∗
z q−d (W, ∗) −→
z q (X, ∗) −→ z q (U, ∗) → z q−d (W, ∗)[1].
If X and W are smooth, we have the distinguished triangle
j∗
i
∗
Z q−d (W, ∗)[−2d] −→
Z q (X, ∗) −→ Z q (U, ∗) → Z q−d (W, ∗)[−2d + 1],
giving the localization sequence for motivic cohomology
(2.8)
j∗
∗
→ H p−2d (W, Z(q − d)) −→
H p (X, Z(q)) −→ H p (U, Z(q))
i
−
→ H p−2d+1 (W, Z(q − d)) →
∂
p
p−2d
The isomorphism iW
(W, Z(q−d)) → HW
(X, Z(q)) resulting from the quasi∗ :H
isomorphism (2.7) is the so-called Gysin isomorphism for motivic cohomology.
2.7. Relative motivic cohomology. We describe a relative version of motivic
cohomology; for simplicity, we restrict ourselves to the affine case.
Let X be a smooth affine k-scheme, essentially of finite type over k, and let
Z1 , . . . , Zn be closed subschemes. For an index I ⊂ {1, . . . , n}, let ZI := ∩i∈I Zi ;
we allow the case I = ∅: Z∅ = X. We suppose that all the ZI are smooth over k.
Let S be a finite set of closed subsets of X, containing all the ZI . Form the
complex Z q (X; Z1 , . . . , Zn , ∗)S as the total complex of the double complex
(2.9)
Z (X, ∗)S →
q
n
i=1
Z q (Zi , ∗)S → . . . →
Z q (ZI , ∗)S → . . .
I⊂{1,... ,n}
|I|=j
with the total degree of Z q (ZI , p)S being p + |I|. Here the differential
Z q (ZI , ∗)S →
Z q (ZI , ∗)S
I⊂{1,... ,n}
|I|=j
I⊂{1,... ,n}
|I|=j+1
THE BLOCH-KATO CONJECTURE AND A THEOREM OF SUSLIN-VOEVODSKY
7
is a signed sum of the pull-back maps
Z q (ZI , ∗)S → Z q (ZI∪{i} , ∗)S ;
i ∈ I,
induced by the inclusions ZI∪{i} ⊂ ZI ; the sign is (−1)l , where l is the number of
j ∈ I with j > i.
The complexes Z q (X; Z1 , . . . , Zn , ∗)S for varying S are all quasi-isomorphic,
since the inclusion (2.3) is a quasi-isomorphism. We will often drop the S from the
notation.
The relative motivic cohomology H p (X; Z1 , . . . , Zn , Z(q)) is defined as
H p (X; Z1 , . . . , Zn , Z(q)) := H p (Z q (X; Z1 , . . . , Zn , ∗));
the mod m version is defined similarly by
H p (X; Z1 , . . . , Zn , Z/m(q)) := H p (Z q (X; Z1 , . . . , Zn , ∗) ⊗ Z/m).
The pull-back maps Z q (ZI , ∗)S → Z q (ZI∪{n} , ∗)S induce the map of complexes
i∗n : Z q (X; Z1 , . . . , Zn−1 , ∗)S → Z q (Zn ; Z1,n , . . . , Zn−1,n , ∗)S
and we have the evident isomorphism
(2.10)
Z q (X; Z1 , . . . , Zn , ∗)S ∼
= cone(i∗n )[−1].
This gives us the long exact relativization sequence
(2.11)
→ H p−1 (Zn ; Z1,n , . . . , Zn−1,n , Z(q)) → H p (X; Z1 , . . . , Zn , Z(q))
i∗
n
H p (Zn ; Z1,n , . . . , Zn−1,n , Z(q)) →
→ H p (X; Z1 , . . . , Zn−1 , Z(q)) −→
and similarly for the mod m version.
2.8. Normal crossing schemes. We describe an extension of the cycle complexes
Z q (X, ∗) to the simplest type of singular schemes, the normal crossing k-schemes.
Taking the component in degree 2q gives an extension to normal crossing schemes
of the notion of a codimension q cycle on a smooth variety.
Let Y be a scheme, essentially of finite type over k, with irreducible components
Y1 , . . . , Yn . For ∅ = I ⊂ {1, . . . , n}, set YI := ∩i∈I Yi . If Y happens to be a closed
subscheme of an essentially smooth k-scheme X, we may consider Y1 , . . . , Yn as
closed subschemes of X, and the notation YI agrees with that given in §2.7.
Definition 2.9. (1) Let Y be essentially of finite type and locally equi-dimensional
over k, with irreducible components Y1 , . . . , Yn . For a point x of Y , let Nx be
the dimension of Y over k at x. We call Y a normal crossing k-scheme if each
intersection YI , I = ∅, is smooth over k, and if, for each point x of Y , a neighborhood
of x in Y is locally isomorphic, in the étale topology, to a union of some coordinate
x +1
hyperplanes in AN
.
k
(2) Let Y be a normal crossing k-scheme, and Z ⊂ Y a closed subscheme. We call
Z a normal crossing subscheme of Y if Z is a normal crossing k-scheme, and Z ∩ YI
is a normal crossing k-scheme for each I = ∅.
For example, a reduced normal crossing divisor D in a smooth k-scheme X is a
normal crossing scheme if each irreducible component of D is smooth over k.
Let Y be a normal crossing k-scheme with irreducible components Y1 , . . . , Yn .
Let S be a finite set of closed subsets of Y , containing all the YI , and let SI ⊂ S
be the set of those closed subsets contained in YI . We assume that for S ∈ S, each
irreducible component of S ∩ YI is also in S; we write Z q (YI , ∗)S for Z q (YI , ∗)SI .
8
THOMAS GEISSER AND MARC LEVINE
Let Z q (Y, ∗)S be the complex defined by the term-wise exactness of
ι
α
(2.12)
→
Z q (Yi , ∗)S −
→
Z q (Yi,j , ∗)S ,
0 → Z q (Y, ∗)S −
1≤i≤n
1≤i<j≤n
where α is the sum of the maps
Z q (Yi , ∗)S → Z q (Yi,j , ∗)S ,
these in turn being the restriction map for i < j, and the negative of the restriction
map for i > j. If we take S to be the set of irreducible components of all the YI ,
we write Z q (Y, ∗) for Z q (Y, ∗)S . We write z q (Y )S for Z q (Y, 2q)S .
Let Y and Y be normal crossing k-schemes, let f : Y → Y be a morphism of kschemes, and take S and S to be the minimal choices, i.e., S is the set of irreducible
components of all the YI , and similarly for S . Assume that the restriction of f to
each YI factors as
g
J
YI −
→ YJ −→
Y
i
for some index J (depending on I), with g flat, and iJ the inclusion. For each i,
choose an index j(i) such that f (Yi ) ⊂ Yj(i)
. Then each of the pull-back maps
∗
q
q
∗
f|Y
:
Z
(Y
,
∗)
→
Z
(Y
,
∗)
is
defined.
Composing f|Y
with the proS
i
S
j(i)
i ,j(i)
i ,j(i)
jection
πj(i) :
Z q (Yj , ∗)S → Z q (Yj(i)
, ∗)S j
gives the map
fi∗ :
Z q (Yj , ∗)S → Z q (Yi , ∗)S .
j
The sum of fi composed with the inclusion ι gives the well-defined pull-back map
f ∗ :=
fi∗ : Z q (Y , ∗) → Z q (Y, ∗).
i
We note that the map f ∗ is independent of the choice of the j(i). Indeed, if
f (Yi ) ⊂ Yj and f (Yi ) ⊂ Yj , then f (Yi ) ⊂ Yj,j
. Letting ρ : Yj,j → Yj , ρ : Yj,j →
Yj be the inclusions, and f¯ : Yi → Yj,j
the map induced by f , we have
∗
∗
◦ πj ◦ ι = f¯∗ ◦ ρ∗ ◦ πj ◦ ι = f¯∗ ◦ ρ∗ ◦ πj ◦ ι = f|Y
f|Y
◦ πj ◦ ι.
i ,j
i ,j
Thus, we have extended the assignment Y → Z q (Y, ∗) to a functor on the category of normal crossing k-schemes, where the morphisms are those morphisms of
k-schemes which admit a factorization as above.
If we enlarge the subset S, we get a pull-back defined for more general maps
than the ones considered above, but it is not clear that the inclusion Z q (Y, ∗)S →
Z q (Y, ∗) is a quasi-isomorphism. The complexes Z q (Y, ∗)S for normal crossing
schemes Y will play only a technical and auxiliary role in our argument, and we will
never require that the cohomology H ∗ (Z q (Y, ∗)S ) is independent of the choice of S.
If, however, Y is essentially smooth and affine, we may use the quasi-isomorphism
(2.3) to give a well-defined pull-back in D(Ab),
(2.13)
as in (2.4).
f ∗ : Z q (Y, ∗) → Z q (Y , ∗),
THE BLOCH-KATO CONJECTURE AND A THEOREM OF SUSLIN-VOEVODSKY
9
Let Z1 , . . . , Zn be closed subschemes of Y . Suppose that all the ZI are normal
crossing subschemes of Y . If S is chosen appropriately, we will have the relative
cycle complex Z q (Y ; Z1 , . . . , Zn )S defined as in §2.7, (2.9), for the smooth case.
2.10. Motivic complexes. The complexes Z q (−, ∗) are functorial for flat maps;
in particular, for an essentially smooth k-scheme X, we may form the complex
of sheaves of abelian groups U → Z q (U, ∗), which we denote by ΓX (q). More
generally, let S be a finite set of closed subsets of X. For U ⊂ X open in X, let
S(U ) = {S∩U |S ∈ S}. Let ΓX (q)S ⊂ ΓX (q) be the Zariski sheaf U → Z q (U, ∗)S(U ) .
If W is a closed subset of X with complement j : V → X, we set
∗
ΓW
X (q)S := cone(j : ΓX (q)S → j∗ ΓV (q)S(V ) )[−1].
Since the inclusion Z q (U, ∗)S(U ) → Z q (U, ∗) is a quasi-isomorphism for U affine
and essentially smooth over k, [14, Chap. II, Theorem 3.5.14], the inclusion
(2.14)
W
ΓW
X (q)S → ΓX (q)
is a quasi-isomorphism for all X essentially smooth over k and all S. Let f : Y → X
be a morphism of essentially smooth k-schemes, and let W ⊂ Y be a closed subset
containing f −1 (W ). We have the stratification S(f ) (2.1), and f ∗ : ΓW
X (q)S →
ΓW
(q)
is
a
well-defined
map
of
complexes
of
sheaves
over
the
map
f
.
Thus we
Y
W
−
have the map f ∗ : ΓW
(q)
→
Rf
Γ
(q)
in
D
(Sh(X
))
defined
by
the
diagram
∗
Zar
Y
X
f∗
W
W
ΓW
X (q) ← ΓX (q)S −→ Rf∗ ΓY (q).
Let Sm/k denote the category of essentially smooth k-schemes. The pull-back
maps f ∗ extend the assignment X → RpX∗ ΓX (q) (pX : X → Spec k the structure
morphism) to a functor
(2.15)
Rp−∗ Γ− (q) : Sm/k op → D− (Ab).
Similarly, let PSm/k denote the category of pairs (X, W ), with X in Sm/k and W a
closed subset of X, where a morphism f : (X , W ) → (X, W ) is a map f : X → X
with f −1 (W ) ⊂ W . Then the assignment (X, W ) → RpX∗ ΓW
X (q) extends to a
functor
(2.16)
op
Rp−∗ Γ−
→ D− (Ab).
− (q) : PSm/k
Let p : X → Spec k be essentially of finite type and locally equi-dimensional over
k. The localization property for the Zariski presheaf of complexes U → Z q (U, ∗)
described in §2.6 implies that the natural map in D− (Ab)
(2.17)
q
ZW
(X, ∗) → Rp∗ ΓW
X (q)
is an isomorphism. Combining the isomorphism (2.17) and the functor (2.16), we
have the functor
(2.18)
q
Z(−)
(−, ∗) : PSm/k op → D− (Ab).
q
(X, W ) → ZW
(W, ∗).
We may sheafify the diagram (8.1) on X × Y , giving the map in D− (Sh(X ×k
YZar ))
(2.19)
∪X,Y
p∗1 ΓX (q) ⊗ p∗2 ΓY (q ) −−−→ ΓX×k Y (q + q ).
10
THOMAS GEISSER AND MARC LEVINE
Via the isomorphism (2.17), this product agrees with the product (8.2). Thus,
by either taking the map on hypercohomology induced by (2.19), or the map on
cohomology induced by (8.2), we have the external product
∪X,Y : H p (X, Z(q)) ⊗ H p (Y, Z(q )) → H p+p (X ×k Y, Z(q + q )).
Reducing all the relevant complexes mod m gives the external cup product
∪X,Y : H p (X, Z/m(q)) ⊗ H p (Y, Z/m(q )) → H p+p (X ×k Y, Z/m(q + q )).
For X essentially smooth, composing ∪X,X with ∆∗ defines the natural cup products
(2.20)
∪X : H p (X, Z(q)) ⊗ H p (X, Z(q )) → H p+p (X, Z(q + q ))
∪X : H p (X, Z/m(q)) ⊗ H p (X, Z/m(q )) → H p+p (X, Z/m(q + q )),
making ⊕p,q H p (X, Z(q)) and ⊕p,q H p (X, Z/m(q)) into bi-graded rings, natural in
X.
3. The construction of cycle maps
We describe the cycle class map from motivic cohomology to étale cohomology,
as well as a relative version. The idea is to apply the cycle class map with values
in étale cohomology with supports (as defined in [5, Cycle]) to the cosimplicial
scheme X × ∆∗ , with support in “codimension q”. The use of “functor complexes”
as described in the next section, is a technical device required for the construction.
3.1. Functor complexes. Let Schk denote the category of k-schemes, essentially
of finite type over k, and let C be a subcategory of Schk .
Suppose we have a functor F : C op → C(Ab). For a cosimplicial scheme in C,
∗
X : ∆ → C, we have the simplicial object n → F (X n ) of C(Ab); we let F (X ∗ )
denote the total complex of the associated double complex:
F (X ∗ )n :=
(3.1)
F p (X −q ).
p+q=n
∗
∗
Sending X to F (X ) defines a contravariant functor from the category of cosimplicial objects of C to C(Ab), natural in F . We have the weakly convergent spectral
sequence
(3.2)
E1p,q = H q (F (X −p )) =⇒ H p+q (F (X ∗ )).
In particular, if φ : F → G is a natural transformation with φ(i) a quasi-isomorphism for each i ∈ C (for short, a quasi-isomorphism), then φ induces an isomorphism
H n (F (X ∗ )) → H n (G(X ∗ )). Also, if X ∗ is a constant cosimplicial scheme with
value X, then the augmentation induces a quasi-isomorphism F (X) → F (X ∗ ).
Let j : U ∗ → X ∗ be an open cosimplicial subscheme in C, and let W ∗ denote
the collection of closed complements W n = X n \ U n ; we call W ∗ a closed subset of
X ∗ . Suppose in addition we have closed cosimplicial subschemes Y1∗ , . . . , Yn∗ of X ∗
(all relevant morphisms being in C). Let F (X ∗ ; Y1∗ , . . . , Yn∗ ) be the total complex
of the double complex
F (X ∗ ) →
n
i=1
F (Yi∗ ) → . . . →
|I|=j
F (YI∗ ) → . . .
THE BLOCH-KATO CONJECTURE AND A THEOREM OF SUSLIN-VOEVODSKY
11
with YI∗ the cosimplicial closed subscheme n → YIn , and the maps in the above
complex defined as in §2.7. We set
(3.3) FW ∗ (X ∗ ; Y1∗ , . . . , Yn∗ ) :=
cone(j ∗ : F (X ∗ ; Y1∗ , . . . , Yn∗ ) → F (U ∗ ; U ∗ ∩ Y1∗ , . . . , U ∗ ∩ Yn∗ ))[−1].
As above, a quasi-isomorphism F → G induces an isomorphism
H ∗ (FW ∗ (X ∗ ; Y1∗ , . . . , Yn∗ )) → H ∗ (GW ∗ (X ∗ ; Y1∗ , . . . , Yn∗ )).
Remark 3.2. A functor F : Schop
k → C(Ab) is called homotopy invariant if the
map
F (X) → F (X × A1 )
induced by the
invariant, then,
p = 0 and E20,q
F (X × ∆∗ ) is a
projection is a quasi-isomorphism for all X. If F is homotopy
for X ∗ = X × ∆∗ , the spectral sequence (3.2) has E2p,q = 0 for
= H q (F (X)). One sees thereby that the augmentation F (X) →
quasi-isomorphism.
Examples 3.3. (1) Let m be an integer with 1/m ∈ k. For a k-scheme X, we
let G∗ (X, µ⊗q
m ) denote the complex of abelian groups gotten by taking the global
sections of the Godement resolution of the étale sheaf µ⊗q
m on X; for example,
0
⊗q
G0 (X, µ⊗q
m ) is the product over all geometric points x of X of Hét (k(x), µm ). Then
∗
⊗q
sending X to G (X, µm ) defines the functor
op
+
G∗ (−, µ⊗q
m ) : Schk → C (Ab).
Since the Godement resolution of µ⊗q
m is a resolution by flasque sheaves, the natural
⊗q
map G∗ (X, µ⊗q
)
→
RΓ(X
,
µ
)
is
an isomorphism in D(Ab); in particular, the
ét
m
m
))
is
canonically
isomorphic to the étale cohomology
cohomology H p (G∗ (X, µ⊗q
m
p
Hét
(X, µ⊗q
).
The
above
discussion
thus
defines
étale cohomology, with coefficients
m
∗
µ⊗q
,
for
a
cosimplicial
scheme
X
,
as
well
as
the
relative version, with support in
m
a closed subset.
(2) Let X be an essentially smooth k-scheme, W ⊂ X a closed subscheme with
complement j : U → X. Let C be the subcategory of SchX with objects the
schemes X × ∆m , U × ∆m , m ≥ 0. The morphisms in C are compositions of maps
of the form j × id and id × g, with g the map induced by a map [m] → [l] in ∆.
Let S be a finite set of closed subsets of X, giving the set S(U ) = {U ∩ S|S ∈ S}.
q
We have the functor z(q)
(−)S : C op → Ab defined by
q
q
z(q)
(X × ∆m )S := z q (X, m)S ; z(q)
(U × ∆m )S := z q (U, m)S(U ) ,
q
with z(q)
(f )S = f ∗ for each morphism f in C. The cycle complex with support in
q
q
W , ZW
(X, ∗)S , of (2.6) is by definition the complex z(q),W
(X × ∆∗ )S [−2q].
(3) Let X be an essentially smooth k-scheme, Z1 , . . . , Zn closed subschemes with
ZI smooth for all I, and S a set of closed subsets of X, containing all the ZI . Let
C be the subcategory of SchX with objects the schemes ZI × ∆m , m ≥ 0. The
morphisms ZI × ∆m → ZJ × ∆l are those of the form i × g, with i an inclusion,
and g the map induced by a map [m] → [l] in ∆.
q
Via the pull-back maps described in §2.3, we have the functor z(q)
(−)S : C op →
Ab defined by
q
z(q)
(ZI × ∆m )S := z q (ZI , m)S ;
12
THOMAS GEISSER AND MARC LEVINE
the relative cycle complex, Z q (X; Z1 , . . . , Zn , ∗)S , of (2.9) is by definition the comq
(X × ∆∗ ; Z1 × ∆∗ , . . . , Zn × ∆∗ )S [−2q].
plex z(q)
(4) For a normal crossing k-scheme Y , with irreducible components Y1 , . . . , Yn , let
S be a set of closed subsets containing all the YI . If we take C to be the subcategory
of SchY formed by the cosimplicial scheme Y × ∆∗ , we form as in (2) the functor
q
z(q)
(−)S : C op → Ab, giving the identity
q
z(q)
(Y × ∆∗ )S [−2q] = Z q (Y, ∗)S ,
where Z q (Y, ∗)S is the cycle complex of (2.12). Similarly, if Z1 , . . . , Zn are closed
subschemes of Y such that each ZI is a normal crossing subscheme of Y , let C be
the category with the objects and morphisms defined as in (3) for the smooth case,
and assume that S contains all the intersections YJ ∩ ZI . We have the functor
zSq : C op → Ab as in (3), and the identity
zSq (Y × ∆∗ ; Z1 × ∆∗ , . . . , Zn × ∆∗ )[−2q] = Z q (Y ; Z1 , . . . , Zn , ∗)S ,
where Z q (Y ; Z1 , . . . , Zn , ∗)S is the relative version of (2.12) defined at the end of
§2.8.
3.4. Cohomology with support. Let k be a field, let X be an essentially smooth
k-scheme, and W a closed subset of X of codimension ≥ q. We denote the group
q
of codimension q cycles on X with support in W by zW
(X). Let m be an integer
prime to the characteristic of k. We recall from [5, Cycle] the construction of the
natural isomorphism
q
2q
cycqW : zW
(X) ⊗ Z/m → HW
(X, µ⊗q
m ),
p
(X, µ⊗q
where HW
m ) is the étale cohomology with support in W .
Suppose at first that X is of finite type over k and W is smooth of codimension
q. By the purity theorem for étale cohomology [19, Chap. VI, Theorem 5.1],
p
∼ 2q−p (W, Z/m);
HW
(X, µ⊗q
m )=H
p
0
in particular, HW
(X, µ⊗q
m ) = 0 for p < 2q. The group H (W, Z/m) is just the free
Z/m-module on theq irreducible components of W , which in turn is isomorphic to
the Z/m-module zW (X) ⊗ Z/m. The construction of [5, Cycles, Chapter 2] gives a
q
2q
canonical choice cycqW of the isomorphism zW
(X) ⊗ Z/m ∼
= HW (X, µ⊗q
m ).
Now suppose that k is perfect and W is an arbitrary closed subset of codimension
≥ q, with singular locus Wsing . By noetherian induction, the localization sequence
for étale cohomology with support [4] gives the isomorphism
2q
⊗q
∼ 2q
HW
(X, µ⊗q
m ) = HW \Wsing (X \ Wsing , µm ),
p
which gives the isomorphism cycqW for general W in X. Similarly, HW
(X, µ⊗q
m )=0
for p < 2q.
The isomorphisms cycqW are compatible with flat pull-back (see e.g. Lemma 3.5
below), hence extend to X essentially smooth over k. We now extend the isomorphisms cycqW to the case of a not necessarily perfect base field k.
Let k0 be the prime field in k; k0 is thus a perfect field. Let X be an essentially smooth k-scheme, and W a codimension ≥ q closed subset. Then the pair
(X, W ) is a filtered projective limit of pairs (Xα , Wα ), where each Xα is a smooth
kα -scheme for some subfield kα of k which is finitely generated over k0 , Wα is a
THE BLOCH-KATO CONJECTURE AND A THEOREM OF SUSLIN-VOEVODSKY
13
closed codimension ≥ q closed subset of Xα , and the transition maps in the projective system are flat and affine. Since motivic cohomology and étale cohomology
with supports transform filtered projective limits with flat affine transition maps
to inductive limits, and since we have compatibility of cycqW with respect to flat
pull-back when defined, it suffices to extend the isomorphisms cycqW to the case in
which k is finitely generated over k0 . In this case, each essentially smooth X over
k is the localization of a smooth k0 -scheme, so the isomorphisms cycqW are defined.
This completes the construction of the isomorphisms cycqW for general base fields;
p
the same argument shows that HW
(X, µ⊗q
m ) = 0 for p < 2q.
Lemma 3.5. (1) The maps cycqW are natural with respect to pull-back: Let f : Y →
X be a morphism of essentially smooth k-schemes, let W ⊂ X be a codimension
≥ q closed subset of X, and let T be a closed subset of Y such that f −1 (W ) ⊂ T
and T has codimension ≥ q on Y . Then
f ∗ ◦ cycqW = cycqT ◦f ∗ .
(2) The maps cycqW are natural with respect to proper push-forward, with the appropriate shift in the codimension: Let f : Y → X be a proper morphism of essentially
smooth k-schemes, of relative dimension d, let W ⊂ X and T ⊂ Y be closed subsets with f (T ) ⊂ W . Suppose that T has codimension ≥ q + d on Y and W has
codimension ≥ q on X. Then
cycqW ◦f∗ = f∗ ◦ cycq+d
.
T
(3) The maps cycqW are compatible with external products: Let X and Y be essentially smooth k-schemes, W ⊂ X and W ⊂ Y closed subsets of codimension ≥ q
q
q
and ≥ q , respectively, and Z ∈ zW
(X), Z ∈ zW
(Y ). Then
p∗1 cycqW (Z) ∪X,Y p∗2 cycqW (Z ) = cycq+q
W ×k W (Z × Z ).
Proof. These properties all follow from [5, Cycles, Chapter 2].
We now extend the definition of the cycle map to normal crossing schemes. Let
Y be a normal crossing scheme with irreducible components Y1 , . . . , Yn . Let W
be a closed subset of Y with complement j : U → Y . For each index J, set
WJ := YJ ∩ W .
Lemma 3.6. Suppose that codimYJ (WJ ) ≥ q for all indices J. Then
p
(1) HW
(Y, µ⊗q
m ) = 0 for p < 2q.
(2) The sequence
2q
0 → HW
(Y, µ⊗q
m )→
2q
HW
(Yj , µ⊗q
→
m )−
j
α
1≤j≤r
2q
HW
(Yj,l , µ⊗q
m )
j,l
1≤j<l≤r
is exact, where α is defined as in §2.7
Proof. For each i = 0, . . . , n, let Y (i) be the disjoint union of the YI with |I| = i,
and let p(i) : Y (i) → Y be the evident morphism. This gives us the standard
resolution of the sheaf µ⊗q
m on Y :
⊗q
⊗q
0 → µ⊗q
m → p∗ µm → . . . → p∗ µm → . . .
(1)
(i)
14
THOMAS GEISSER AND MARC LEVINE
(to see that this is indeed a resolution, reduce to the case of the union of n hyperplanes in AN +1 , and use induction on n and N ). The corresponding hypercohomology spectral sequence gives the Mayer-Vietoris spectral sequence
a+b
b
⊗q
E1a,b =
HW
(YJ , µ⊗q
m ) =⇒ HW (Y, µm ).
J
|J|=a+1
E1a,b
By purity, we have
= 0 for b < 2q; since E1a,b = 0 for a < 0, E1a,b = 0 for
a + b < 2q, whence (1). Similarly, the only non-zero terms with a + b ≤ 2q + 1 and
b ≤ 2q are E10,2q , E11,2q . Thus
2q
0,2q
E20,2q = E∞
= HW
(Y, µ⊗q
m ).
The map α is d0,2q
, proving (2).
1
Recall from §2.8 the cycle complex Z q (Y, ∗)S , and the group of cycles on Y ,
z q (Y )S := Z q (Y, 2q)S . Explicitly, z q (Y )S is the kernel of the map
α
z q (Yi )S −
→
z q (Yi,j )S
1≤i≤n
1≤i<j≤n
q
with α as in §2.8. If W is a closed subset of Y , we let zW
(Y )S be the subgroup of
q
z (Y )S consisting of cycles with support in W . We omit the S from the notation
if S is the minimal choice, i.e., the set of all the YI .
We have the exact sequence
q
q
q
0 → zW
(3.4)
(Y )S →
zWi (Yi )S →
zW
(Yi,j )S ,
i,j
1≤i≤n
1≤i<j≤n
with the maps defined as above. If we suppose that W satisfies the conditions of
Lemma 3.6, there therefore is a unique homomorphism
q
2q
cycqW : zW
(Y ) → HW
(Y, µ⊗q
m )
which makes the diagram
/
q
zW
(Y )
q
zW
(Yi )
i
1≤i≤n
cycqW
2q
HW
(Y, µ⊗q
m )
/
⊕ cycqW
i
2q
HW
(Yi , µ⊗q
m )
i
1≤i≤n
commute. Using Lemma 3.5(1) we see that the cycqW are functorial for maps of normal crossing schemes f : (Y , W ) → (Y, W ) (satisfying the factorization conditions
of §2.8) in case W ⊃ f −1 (W ) and W satisfies the conditions of Lemma 3.6.
3.7. The cycle class map. We first construct the cycle class map as a map
clq : Z(X, ∗)S ⊗L Z/m → G∗ (X, µ⊗q
m )
in D(Ab), where X is an essentially smooth k-scheme, S is a finite set of closed
subsets of X, and G∗ (X, µ⊗q
m ) is the Godement resolution of Example 3.3(1). We
then extend this to a version for the relative cycle complexes for a normal crossing
k-scheme. In the next section, we give a sheafified version of clq , which, among
other things, enables us to show that clq is natural.
THE BLOCH-KATO CONJECTURE AND A THEOREM OF SUSLIN-VOEVODSKY
15
For a category C, we let Func(C, Ab) denote the abelian category of functors
C → Ab. We may also form the category of complexes C∗ (Func(C, Ab)) (∗ a
boundedness condition), and the derived category D∗ (Func(C, Ab)).
Let X be a reduced k-scheme, and S a finite set of closed subsets of X, with
X ∈ S. Let S0 be a subset of S consisting of (reduced) schemes locally equidimensional over k. Let C(S0 ) be the subcategory of SchX with objects the schemes
of the form S × ∆m , with S ∈ S0 , where a morphism S × ∆m → T × ∆l is a map of
the form i × g, with i : S → T an inclusion, and with g : ∆m → ∆l the affine-linear
map induced from some ḡ : [m] → [l] in ∆.
For example, when we use C(S0 ) to define the cycle class map for the relative
complex Z q (X; Z1 , . . . , Zn , ∗)S , we take S0 to be the set of irreducible components
of the ZI . The set S is needed to show the naturality of the cycle class map.
(q)
For S ∈ S0 , let (S × ∆m )S be the set of closed subsets W of S × ∆m such that,
for each T ∈ S, T ⊂ S, and each face F of ∆m , the intersection W ∩ (T × F ) has
codimension ≥ q on T × F . We set
G∗(q) (S × ∆m , µ⊗q
m )S :=
G∗W (S × ∆m , µ⊗q
m ).
lim
→
(q)
W ∈(S×∆m )S
Sending S × ∆m to G∗(q) (S × ∆m , µ⊗q
m )S defines the functor
op
G∗(q) (−, µ⊗q
→ C(Ab).
m )S : C(S0 )
q
Assume that each S ∈ S0 is a normal crossing scheme. Let z(q)
(−)S : C(S0 )op →
Ab be the functor
q
z(q)
(S × ∆m )S =
lim
→
q
zW
(S × ∆m ),
(q)
W ∈(S×∆m )S
q
z(q)
(f )S
= f ∗;
the conditions on S and W imply that the relevant cycle intersection products
q
are all defined, so that z(q)
(−)S is well-defined. In addition, for each S ∈ S0 ,
q
the restriction of z(q) (−)S to the subcategory C({S})op of C(S0 )op agrees with the
q
functor z(q)
(−)S defined in Example 3.3(4), i.e.,
q
z(q)
(S × ∆m )S = z q (S, m)S .
(3.5)
As noted in the Examples 3.3(2)-(4), we have
(3.6)
q
z(q)
(S × ∆∗ )S [−2q] = Z q (S, ∗)S
for each S ∈ S0 .
Lemma 3.8. Suppose that
1. Each S ∈ S0 is a normal crossing scheme.
2. If S ∈ S0 has irreducible components S1 , . . . , Sm , then each SJ is in S0 .
Then there is a unique isomorphism in Func(C(S0 )op , Ab),
q
cycq(q) : z(q)
(−)S ⊗ Z/m → H 2q (G∗(q) (−, µ⊗q
m )S ),
16
THOMAS GEISSER AND MARC LEVINE
(q)
such that, for S ∈ S0 smooth, and W ∈ (S × ∆m )S , the diagram
q
zW
(S × ∆m )S ⊗ Z/m
q
z(q)
(S × ∆m )S ⊗ Z/m
cycqW
cycq(q)
/ H 2q (G∗ (S × ∆m , µ⊗q )S )
m
W
/ H 2q (G∗(q) (S × ∆m , µ⊗q
m )S )
commutes. In addition, we have
H p (G∗(q) (−, µ⊗q
m )S ) = 0
for p < 2q.
q
q
Proof. Since z(q)
(S × ∆m )S is the direct limit of the zW
(S × ∆m ), the uniqueness
q
of cyc(q) is clear. The existence follows directly from Lemma 3.5, Lemma 3.6 and
the comments following Lemma 3.6; the cohomology vanishing for p < 2q follows
from Lemma 3.6.
We apply the lemma to construct the cycle class map from motivic cohomology
to étale cohomology. We first illustrate the most basic case, that of an essentially
smooth k-scheme X. Take S0 = {X}. We have the diagram
(3.7)
q
z(q)
(−)S [−2q] ⊗ Z/m
cycq(q) [−2q]
/ H 2q (G∗(q) (−, µ⊗q
m )S )[−2q]
O
α
o
G∗ (−, µ⊗q
m )
τ≤2q G∗(q) (−, µ⊗q
m )S ,
β
in C(Func(C({X})op , Ab)), where α is the canonical map, and β is the composition
∗
⊗q
of the canonical inclusion τ≤2q G∗(q) (−, µ⊗q
m )S → G(q) (−, µm )S , followed by the
∗
⊗q
∗
⊗q
“forget the support” map G(q) (−, µm )S → G (−, µm ). By Lemma 3.8, the map
α is a quasi-isomorphism. Thus, the diagram (3.7) defines a map
(3.8)
q
z(q)
(−)S [−2q] ⊗ Z/m → G∗ (−, µ⊗q
m )
in D(Func(C({X})op , Ab)).
We now apply the map (3.8) to the cosimplicial scheme X × ∆∗ , in the sense of
§3.1. As noted in (3.6), we have
q
z(q)
(X × ∆∗ )S [−2q] = Z q (X, ∗)S .
By Remark 3.2, and the homotopy property for étale cohomology, the augmentation
∗
∗
⊗q
to X induces a quasi-isomorphism G∗ (X, µ⊗q
m ) → G (X × ∆ , µm ). Thus, we have
constructed the map in D(Ab)
(3.9)
clq : Z q (X, ∗)S ⊗ Z/m → G∗ (X, µ⊗q
m ).
From the construction of clq , it is obvious that, for S ⊂ S, the diagram
(3.10)
/ Z q (X, ∗)S ⊗ Z/m
Z q (X, ∗)S ⊗ Z/m
SSS
SSS
SSS
clq
SSS
clq
SS)
G∗ (X, µ⊗q
m )
i
THE BLOCH-KATO CONJECTURE AND A THEOREM OF SUSLIN-VOEVODSKY
17
commutes, where i is the evident inclusion of complexes. Taking S = {X} and
taking cohomology, we have the mod m cycle class map
(3.11)
∗
clq : H ∗ (X, Z/m(q)) → Hét
(X, µ⊗q
m ).
Remark 3.9. Assuming X affine, we may enlarge the set S without changing the
cohomology of Z q (X, ∗)S , since the map (2.3) is a quasi-isomorphism. From this,
one easily sees that clq is natural for affine X. We will discuss the naturality in
general in the next section.
We may modify the construction by restricting the support throughout to a fixed
closed subset W of X. This gives the version with support in W ,
(3.12)
q
clq : ZW
(X, ∗)S ⊗ Z/m → G∗W (X, µ⊗q
m ),
and on taking cohomology
(3.13)
∗
∗
clq : HW
(X, Z/m(q)) → HW
(X, µ⊗q
m ).
A similar construction gives an extension of the cycle class map to the setting
of relative cohomology of normal crossing schemes (including the case of relative
cohomology for smooth schemes). Indeed, let Y be a normal crossing scheme with
irreducible components Y1 , . . . , Ym , and let Z1 , . . . , Zn closed normal crossing subschemes of Y such that each ZI is a closed normal crossing subscheme of Y . Let S0
be the set of irreducible components of all the ZI ∩YJ , and let S ⊃ S0 be a finite set
of closed subsets of Y . One easily sees that S0 satisfies the conditions of Lemma 3.8.
We have the diagram (3.7) in C(Func(C(S0 )op , Ab)), with α a quasi-isomorphism.
We thus get a similar diagram in C(Ab) upon evaluation (in the sense of §3.1) at
(Y × ∆∗ ; Z1 × ∆∗ , . . . , Zn × ∆∗ ). The homotopy property for étale cohomology
gives, as above, the quasi-isomorphism
∗
∗
∗
∗
⊗q
G∗ (Y ; Z1 , . . . , Zn , µ⊗q
m ) → G (Y × ∆ ; Z1 × ∆ , . . . , Zn × ∆ , µm ).
Using the identity (3.6), we have
q
z(q)
(Y × ∆∗ ; Z1 × ∆∗ , . . . , Zn × ∆∗ )S [−2q] = Z q (Y ; Z1 , . . . , Zn , ∗)S ,
giving the map clq : Z q (Y ; Z1 , . . . , Zn )S → G∗ (Y ; Z1 , . . . , Zn , µ⊗q
m ) in D(Ab). If
we take S to be the minimal choice, or if we assume that Y and all the ZI are
smooth and affine, taking cohomology gives us the cycle class map
(3.14)
∗
clq : H ∗ (Y ; Z1 , . . . , Zn , Z/m(q)) → Hét
(Y ; Z1 , . . . , Zn , µ⊗q
m ).
3.10. Cycle classes for motivic complexes. We give a refinement of the cohomological cycle class map described above to a natural map in the derived category
D− (Sh(XZar )) of Zariski sheaves on an essentially smooth k-scheme X, as well as
a variant for complexes with support. As a tool, we construct a category C/k out
of pairs consisting of an essentially smooth k-scheme X and a finite set of closed
subsets of X, so that the operation of taking the codimension q cycles on (X, S)
becomes functorial for pull-back.
As in §3.7, let X be an essentially smooth k-scheme, and let S be a finite set of
closed subsets of X, with X ∈ S. For S ∈ S, we let S∗ be the union of the S ∈ S
which contain no generic point of S, and let S 0 denote the open subset S \ S∗ of S.
Let C/k be the following category: Objects are pairs (T, S(T )) consisting of an
essentially smooth k-scheme T , and a finite set of closed subsets of T , S(T ). A
18
THOMAS GEISSER AND MARC LEVINE
morphism (T, S(T )) → (T , S(T )) in C/k is a morphism of k-schemes, g : T → T ,
such that such that the following condition holds:
(3.15)
For each S ∈ S(T ), there is an S ∈ S(T ) such that S 0 ⊂ f −1 (S 0 ) and f :
0
S → S 0 is equi-dimensional.
Recall the category C(S0 ) defined in §3.7; in particular, for an essentially smooth
k-scheme T , we have the category C({T }). Let C({∗}) be the category with
Obj C({∗}) :=
Obj C({T }).
(T,S(T ))∈C/k
For T × ∆ in the component (T , S(T )) and T × ∆l in the component (T, S(T )),
a morphism T ×∆m → T ×∆l is a map of k-schemes of the form f ×g : T ×∆m →
T ×∆l , with f : (T, S(T )) → (T , S(T )) a morphism in C/k, and idT ×g a morphism
in C({T }). Sending T × ∆m to (T, S(T )) and f × g to f makes C({∗}) a category
over C/k, with fiber C({T }) over (T, S(T )).
For each (T, S(T )) ∈ C/k, we have the functors defined in §3.7
m
q
op
z(q)
(−)S(T ) , G∗(q) (−, µ⊗q
→ C(Ab).
m )S(T ) : C({T })
For each f : (T , S(T )) → (T, S(T )) in C/k and for each W ∈ (T × ∆n )S(T ) ,
the condition (3.15) implies that the cycle (f × id)∗ (W ) is defined and is in (T ×
(q)
∆n )S(T ) . Thus the assignments
(q)
q
T × ∆n → z(q)
(T × ∆n )S(T )
T × ∆n → G∗(q) (T × ∆n , µ⊗q
m )S(T )
extend to functors
q
op
z(q)
(−)S , G∗(q) (−, µ⊗q
→ C(Ab).
m )S : C({∗})
The construction of §3.7 yields the diagram (3.7) in C(Func(C({∗})op , Ab)), and
the natural map
(3.16)
q
z(q)
(−)S [−2q] ⊗ Z/m → G∗ (−, µ⊗q
m )
in D(Func(C({∗})op , Ab)). Applying (3.16) to T ×∆∗ for (T, S(T )) in C/k and com∗
∗
⊗q
posing with the inverse of the quasi-isomorphism G∗ (T, µ⊗q
m ) → G (T × ∆ , µm )
gives us the natural map in D(Func(C/k op , Ab)),
(3.17)
clq : Z (q) (−, ∗)S ⊗ Z/m → G∗ (−, µ⊗q
m ),
where Z (q) (−, ∗)S is the functor (T, S(T )) → Z (q) (T, ∗)S(T ) .
Call a map f : (T , S(T )) → (T, S(T )) étale if the underlying map of schemes
f : T → T is étale, and if S(T ) = f −1 (S(T )). We make C/k into a site C/két by
defining a covering family of (T, S(T )) to be a collection of étale maps such that
the underlying family of maps of schemes is an étale cover. The Zariski site C/kZar
is defined analogously. Clearly, for each (T, S(T )), the restriction of C/két (resp.
C/kZar ) to (T, S(T )) is isomorphic to the small étale site Tét (resp. small Zariski
site TZar ).
Let ˆ : C/két → C/kZar be the change of topology map. For each essen+
tially smooth k-scheme T , G∗ (T, µ⊗q
m ) is a functorial representative in C (Ab)
THE BLOCH-KATO CONJECTURE AND A THEOREM OF SUSLIN-VOEVODSKY
19
⊗q
of RΓ(Tét , µ⊗q
m ). Letting G(µm )T denote the complex of Zariski sheaves associated
to the presheaf
(U ⊂ T ) → G∗ (U, µ⊗q
m ),
it follows that T → G(µ⊗q
∗ µ⊗q
m )T is a functorial complex of sheaves representing Rˆ
m .
The map (3.17), together with the quasi-isomorphism (2.14), thus gives us the
map in D(Sh(C/kZar ))
(3.18)
clq : Γ(−) (q) ⊗L Z/m → Rˆ
∗ (µ⊗q
m ),
The natural map τ≤q ΓX (q) → ΓX (q) is a quasi-isomorphism, since H p (R, Z(q)) =
0 if p > q, for a regular local k-algebra R. Thus (3.18) defines the map in
D− (Sh(C/kZar ))
(3.19)
clq : Γ(−) (q) ⊗L Z/m → τ≤q Rˆ
∗ (µ⊗q
m ).
If we evaluate (3.19) at (X, {X}), we obtain the map in D− (Sh(XZar ))
(3.20)
clq : ΓX (q) ⊗L Z/m → τ≤q RX∗ (µ⊗q
m ),
giving us the sheafified version of (3.9).
Similarly, if W is a closed subset of X with complement j : U → X, we may
apply (3.19) to the map j (i.e., take cone(j ∗ )[−1]), giving the map for the complexes
with support
(3.21)
L
!
⊗q
clq : ΓW
X (q) ⊗ Z/m → RX∗ RiW (µm ).
4. Properties of the cycle class map
4.1. Naturality of the cycle class map. We proceed to show that the cycle
class maps defined in the previous section are natural. For this, we show that the
sheaf-theoretic cycle class map (3.21) is natural, and that it gives rise to the “naive”
cycle class maps defined in §3.7 by passing to hypercohomology.
Recall the category of pairs PSm/k defined in §2.10. Let (X, W ) be in PSm/k.
The natural isomorphism RΓ(XZar , −) ◦ R∗ ∼
= RΓ(Xét , −) gives us the natural iso!
⊗q
)
→
RΓ(X
morphism η : RΓ(XZar , RX∗ Ri!W µ⊗q
ét , RiW µm ). We have the natural
m
isomorphisms
q
φ : ZW
(X, ∗) → RΓ(XZar , ΓW
X (q)),
!
⊗q
φét : G∗W (X, µ⊗q
m ) → RΓ(Xét , RiW µm )
in D(Ab), the first one being the isomorphism (2.17), and the second coming from
the fact that the functor Y → G∗ (Y, µ⊗q
m ) of Example 3.3(1) represents the functor
op
Y → RΓ(Yét , µ⊗q
m ) from Schk to D(Ab).
Thus, the construction of clq gives us the commutative diagram in D(Ab)
(4.1)
q
ZW
(X, ∗) ⊗L Z/m
φ
∼
η◦RΓ(clqX )
clq
G∗W (X, µ⊗q
m )
/ RΓ(XZar , ΓW (q) ⊗L Z/m)
X
∼
φét
/ RΓ(Xét , Ri! µ⊗q ).
W m
The cycle class map (3.13) for X with supports in W is thus the map on H∗ (XZar , −)
induced by the sheafified version (3.21).
This discussion gives the naturality of the cycle class maps defined in the previous
section.
20
THOMAS GEISSER AND MARC LEVINE
Proposition 4.2. (1) The cycle class maps (3.9), (3.11) and (3.20) are natural
on Sm/k.
(2) The cycle class maps with support (3.12), (3.13) and (3.21) are natural on
PSm/k.
(3) The cycle class map for relative cohomology (3.14) is natural on the category
of tuples (X, Z1 , . . . , Zn ) of essentially smooth affine k-schemes X, with closed
subschemes Zj such that each ZI is essentially smooth over k, where a morphism
f : (X , Z1 , . . . , Zn ) → (X, Z1 , . . . , Zn )
is a morphism f : X → X such that f (Zi ) ⊂ Zi for all i.
(4) The cycle class map for relative cohomology (3.14) is natural on the category
of tuples (Y, Z1 , . . . , Zn ) of normal crossing k-schemes Y , with closed subschemes
Zj such that each ZI is a normal crossing subscheme of Y , where the morphisms
are as described in §2.8.
(5) The cycle class map (3.14) (for n = 0) is natural for arbitrary maps f : Y → X,
where X is affine and essentially smooth over k, Y is a normal crossing k-scheme,
and f ∗ : Z q (X, ∗) → Z q (Y, ∗) is the map (2.13).
Proof. (1) is clearly a special case of (2). For (2), let f : (X , W ) → (X, W ) be a
morphism in PSm/k. We have the set S(f ) (2.1) of closed subsets of X, giving us
the lifting of the commutative diagram in Sm/k
j
X \ W / X
f
f
X \W
/X
j
to the commutative diagram in C/k
(X \ W , {X \ W })
j
f
(X \ W, S(f|X \W ))
/ (X , {X })
f
j
/ (X, S(f )).
Applying the morphism (3.19) to this diagram, dropping the truncation, and taking
the cones with respect to the horizontal maps gives the commutative diagram in
D(Sh(XZar ))
ΓW
X
f∗
clq
RX∗ Ri!W µ⊗q
m
f∗
/ Rf∗ ΓW
X
Rf∗ clq
/ Rf∗ RX ∗ Ri! µ⊗q .
W m
Applying RΓ(XZar , −) to this diagram and using the commutativity of the diagram
(4.1) proves (2).
THE BLOCH-KATO CONJECTURE AND A THEOREM OF SUSLIN-VOEVODSKY
21
For (3), the naturality of the diagram (3.7) gives the commutativity of the diagram
Z q (X; Z1 , . . . , Zn , ∗)
VVVV
O
VVVVclq
VVVV
id∗
VVVV
V*
clq /
q
Z (X; Z1 , . . . , Zn , ∗)S(f )
G∗ (X; Z1 , . . . , Zn , µ⊗q
m )
f∗
Z q (X ; Z1 , . . . , Zn , ∗)
cl
q
f∗
/ G∗ (X ; Z , . . . , Z , µ⊗q )
n
m
1
in D(Ab), which gives the desired naturality on taking cohomology. The proof of
(4) and (5) are similar, and are left to the reader.
4.3. Compatibility with Gysin maps. We prove that the maps clq are compatible with the localization/Gysin sequences for motivic cohomology/étale cohomology. For a scheme X, we let Sm/X denote the category of schemes smooth and
essentially of finite type over X.
Let i : Y → X be a smooth codimension d closed subscheme of an essentially
smooth k-scheme X. Let T → X be a smooth X-scheme, and let TY → Y denote
the fiber product T ×X Y , with inclusion iT : TY → T . The canonical Gysin
!
⊗q+d
∼
isomorphisms µ⊗q
[2d] in D(Sh(TY ét )) (see [19, VI, Theorem 6.1])
m = RiTY µm
give rise to the isomorphism in D(Func(Sm/X op , Ab))
∗
⊗q+d
ι : [T → G∗ (TY , µ⊗q
)[2d] .
m )] → T → GTY (T, µm
Let iW : W → TY be a closed subset of TY , with complement jY : VY → TY in
TY and j : V → T in T . Evaluating ι at T and V gives the canonical map of cones
j∗
Y
∗
⊗q
cone[G∗ (TY , µ⊗q
m ) −→ G (VY , µm )][−1]
j∗
ιW (T )
−−−−→ cone[G∗TY (T, µ⊗q+d
) −→ G∗VY (V, µ⊗q+d
)][2d − 1].
m
m
Since W = TY \ VY = T \ V , it follows that
j∗
Y
∗
⊗q
∗
⊗q
cone[G∗ (TY , µ⊗q
m ) −→ G (VY , µm )][−1] = GW (TY , µm ),
and that the evident map
j∗
cone[G∗TY (T, µ⊗q+d
) −→ G∗VY (V, µ⊗q+d
)][−1] → G∗W (T, µ⊗q+r
)
m
m
m
∗
⊗q+r
is a homotopy equivalence. In addition, G∗W (TY , µ⊗q
) are comm ) and GW (T, µm
! ⊗q+d
))
and
RΓ(T,
R(i
◦i
)
(µ
)),
respectively.
plexes representing RΓ(TY , Ri!W (µ⊗q
T W
m
m
Thus the map
(4.2)
∗
⊗q+r
ιW (T ) : G∗W (TY , µ⊗q
)[2d]
m ) → GW (T, µm
represents the natural isomorphism in D(Ab)
(4.3)
! ⊗q+d
ιW : RΓ(TY , Ri!W (µ⊗q
)[2d]),
m )) → RΓ(T, R(iT ◦ iW ) (µm
p
∼ p+2d (T, µ⊗q+d ) on
(TY , µ⊗q
which in turn induces the Gysin isomorphism HW
m ) = HW
m
taking cohomology.
22
THOMAS GEISSER AND MARC LEVINE
Lemma 4.4. Let i : Y → X be a closed codimension d embedding of essentially
smooth k-schemes. Then the diagram in D− (Sh(XZar ))
clqY
ΓY (q)
α
ΓYX (q + d)[2d]
/ RY ∗ µ⊗q
m
β
clq+d
X
/ RX∗ Ri! (µ⊗q+d )[2d]
m
Y
commutes, where α and β are the respective Gysin isomorphisms.
Proof. Let U be an open subscheme of X, UY = Y ∩ U . Applying the Gysin
isomorphism (4.3) for T = U × ∆m , and for W in (UY × ∆m )(q) , we have the
following diagram in D(Ab):
(4.4)
q
zW
(UY × ∆m ) ⊗ Z/m[−2q]
cycq [−2q]
q+d
zW
(U × ∆m ) ⊗ Z/m[−2q]
cycq+d [−2q]
ι1
2q
HW
(UY × ∆m , µ⊗q
m )[−2q]
O
p
p
τ≤2q G∗W (UY × ∆m , µ⊗q
m )
ι2
ι3
p∗
1
G∗ (UY , µ⊗q
m )
/ τ≤2q G∗ (U × ∆m , µ⊗q+d
)[2d]
m
W
q
q
G∗ (UY × ∆m , µ⊗q
m )
O
/ H 2(q+d) (U × ∆m , µ⊗q+d )[−2q]
m
W
O
m
⊗q+d
/ G∗
)[2d]
UY ×∆m (U × ∆ , µm
O
p∗
1
ι4
/ G∗ (U, µ⊗q+d
)[2d]
m
UY
Here the left-hand column is the diagram used in defining clq in §3.7, the right-hand
column is the diagram used in defining the version of clq for U with supports in UY ,
and the maps ιi are induced by the appropriate Gysin map (4.2). In particular, all
the vertical maps, except for q and q , are quasi-isomorphisms, and all the horizontal
maps are isomorphisms in D(Ab). It follows from Lemma 3.5 that the top square
commutes; the next two squares commute in D(Ab) since ι2 = τ≤2q ι3 by definition,
and ι1 is the map on cohomology induced by ι2 . The bottom square commutes in
D(Ab) by the naturality of the Gysin isomorphism.
Since the ιi come from isomorphisms in D(Func(Sm/X op , Ab)), (4.4) defines a
commutative diagram in D(Func(Sm/X op , Ab)). Taking the limit over W , this
THE BLOCH-KATO CONJECTURE AND A THEOREM OF SUSLIN-VOEVODSKY
23
gives us the commutative diagram in the derived category of presheaves on X:


ι
/ z q+d (U, ∗) ⊗ Z/m[−2q]
z q (UY , ∗) ⊗ Z/m[−2q]
UY






q
q+d
cyc
cyc






ι
∗
∗
⊗q+d
/
(4.5)
U →  G∗ (UY × ∆∗ , µ⊗q
GUY ×∆∗ (U × ∆ , µm )[2d] 
m )


O
O




∗
∗
p1
p1




ι
/ G∗ (U, µ⊗q+d
)[2d]
G∗ (UY , µ⊗q
)
m
UY
m
As clq = (p∗1 )−1 ◦cycq and clq+d = (p∗1 )−1 ◦cycq+d , sheafifying (4.5) gives the desired
commutative diagram in D− (Sh(XZar )).
Proposition 4.5. (1) The cycle class map for relative cohomology is compatible
with the maps in the long exact relativization sequences for motivic cohomology and
étale cohomology.
(2) Let i : Y → X be a closed codimension d embedding of essentially smooth
k-schemes with complement j : U → X. Then the diagram
H p−1 (U, Z/m(q))
∂
clq
p−1
Hét
(U, µ⊗q
m )
/ H p−2d (Y, Z/m(q − d))
i∗
clq
clq−d
∂
/ H p−2d (Y, µ⊗q−d )
m
ét
/ H p (X, Z/m(q))
i∗
/ H p (X, µ⊗q
m )
ét
commutes.
Proof. (1) follows immediately from the naturality of the map (3.8) and the definition of relative motivic cohomology (resp. relative étale cohomology) via cones.
For (2), the same reasoning as in (1) gives us the map of distinguished triangles
in D(Ab):
/ Z q (X, ∗)
ZYq (X, ∗)
clq
j∗
clq
G∗Y (X, µqm )
/ G∗ (X, µqm )
/ Z q (U, ∗)
clq
j
∗
/ G∗ (U, µqm )
which in turn gives the commutative diagram
(4.6)
H p−1 (U, Z/m(q))
∂
clq
p−1
Hét
(U, µ⊗q
m )
/ H p (X, Z/m(q))
Y
i∗
clq
∂
/ H p (X, µ⊗q
m )
Y
/ H p (X, Z/m(q))
clq
i∗
/ H p (X, µ⊗q
m )
ét
Since the diagram in (2) is derived from (4.6) by replacing HYp (X, Z/m(q)) and
p−2d
p−2d
HYp (X, µ⊗q
(Y, Z/m(q − d)) and Hét
(Y, µ⊗q−d
) via the respective
m ) with H
m
Gysin isomorphisms, Lemma 4.4 together with the commutativity of (4.6) proves
(2).
24
THOMAS GEISSER AND MARC LEVINE
4.6. Products. We proceed to check that the cycle class map is compatible with
products.
Proposition 4.7. Let X and Y be k-schemes, of finite type and locally equidimensional over k. The diagram
H p (X, Z/m(q)) ⊗ H p (Y, Z/m(q ))
clq ⊗ clq
/ H p+p (X ×k Y, Z/m(q + q ))
∪
p
p
Hét
(X, µ⊗q
)
⊗
Hét
(Y, µ⊗q
m )
m
clq+q
/ H p+p (X × Y, µ⊗q+q )
k
m
ét
∪
is commutative. In particular, for X essentially smooth over k, the map
p
cl∗ : ⊕p,q H p (X, Z/m(q)) → ⊕p,q Hét
(X, µ⊗q
m )
is a ring homomorphism.
Proof. Following [9, Chapter 6], a map of sheaves µ : F ⊗ G → H on a topological
space X induces a functorial map (over µ) on the Godement resolutions
G(µ) : G∗ (X, F) ⊗ G∗ (X, G) → G∗ (X, H).
We apply the methods of §3.7, which, together with Lemma 3.5(3), gives us the
commutative diagram in D− (ModZ/m )
∪
q
Z q (X, ∗)/m ⊗L
Z/m Z (Y, ∗)/m
clq ⊗ clq
/ Z q+q (X × Y, ∗)/m
clq+q
L
∗
⊗q
G∗ (X, µ⊗q
m ) ⊗Z/m G (Y, µm )
∪
/ G∗ (X × Y, µ⊗q+q )
m
Here, the map ∪ is defined by the diagram analogous to (8.1):
p∗ ⊗p∗
1
1
L
∗
⊗q
∗
∗
⊗q
L
∗
∗
⊗q
G∗ (X, µ⊗q
m ) ⊗ G (Y, µm ) −−−−→ G (X × ∆ , µm ) ⊗ G (Y × ∆ , µm )
G( )
−−−→ G∗ (X × Y × ∆∗ × ∆∗ , µ⊗q+q
)
m
−
→ G∗ (X × Y × ∆∗ , µ⊗q+q
)←
− G∗ (X × Y, µ⊗q+q
).
m
m
T
It thus suffices to see that ∪ agrees with the product
L
∗
⊗q
∗
⊗q+q
∪ : G∗ (X, µ⊗q
)
m ) ⊗Z/m G (Y, µm ) → G (X × Y, µm
⊗q
⊗q+q
induced by the product of sheaves µ⊗q
. Since the augmentations
m ⊗ µm → µm
∗
∗
⊗q
G∗ (X, µ⊗q
m ) → G (X × ∆ , µm )
∗
∗
⊗q
G∗ (Y, µ⊗q
m ) → G (Y × ∆ , µm )
G∗ (X × Y, µ⊗q+q
) → G∗ (X × Y × ∆∗ , µ⊗q+q
)
m
m
are quasi-isomorphisms, this follows from the fact that the map
G∗ (X × Y × ∆∗ × ∆∗ , µ⊗q+q
)−
→ G∗ (X × Y × ∆∗ , µ⊗q+q
)
m
m
T
is a map over the identity on G∗ (X × Y, µ⊗q+q
).
m
THE BLOCH-KATO CONJECTURE AND A THEOREM OF SUSLIN-VOEVODSKY
25
4.8. The case of fields. We discuss some additional properties of the cycle class
map for X = Spec F , F a field.
From [1, Theorem 6.1], the map sending u ∈ F \ {0, 1} to the cycle [u] :=
1
u
1 · ( 1−u
, u−1
) of ∆1F gives rise to an isomorphism F ∗ → H 1 (F, Z(1)). Additionally, H p (F, Z(1)) = 0 for p = 1, hence we have the isomorphism F ∗ /F ∗m ∼
=
1
H 1 (F, Z/m(1)). The Kummer sequence gives Hét
(F, µm ) ∼
= F ∗ /F ∗m for all m
prime to char F .
Proposition 4.9. Let u be in F \ {0, 1}. The cycle class map
cl1 : H 1 (F, Z/m(1)) → H 1 (F, µm ) ∼
= F ∗ /F ∗m
ét
sends the class of [u] to the class of u mod F ∗m , hence is an isomorphism.
.
Proof. Let U = ∆1F \ {[u]}. We have the Gysin sequence
∂
1
2
2
(U, µm ) −
→ H[u]
(∆1F , µm ) → Hét
(∆1F , µm ).
Hét
By definition of the étale cycle class of a divisor [5, Cycle, Définition 2.1.2], it follows
2
that the cycle class of [u] in H[u]
(∆1F , µm ) is given by cyc1[u] ([u]) = ∂([f ]), where
1
1
f is a regular function on ∆F with div f = [u], and [f ] is the class in Hét
(U, µm )
0
corresponding to f ∈ Hét (U, Gm ) via the Kummer sequence.
We claim that
(4.7)
1
cl1 ([u]) = i∗(1,0) ([f ]) − i∗(0,1) ([f ]) ∈ Hét
(F, µm ).
To see this, let {[f ]} be a cocycle in G1 (U, µm ) representing [f ]. Let G̃p[u] (∆1F , µm )
j∗
be the kernel of the surjection Gp (∆1F , µm ) −→ Gp (U, µm ), giving the term-wise
exact sequence of complexes
(4.8)
j∗
∗
0 → G̃∗[u] (∆1F , µm ) −→
G∗ (∆1F , µm ) −→ G∗ (U, µm ) → 0.
i
The canonical map G̃∗[u] (∆1F , µm ) → G∗[u] (∆1F , µm ) is a homotopy equivalence, and
the Gysin sequence for the inclusion of [u] in ∆1F is the long exact cohomology
sequence resulting from (4.8). Thus, if φ ∈ G1 (∆1F , µm ) is a cochain lifting {[f ]},
then dφ = i∗ η, η ∈ G̃2[u] (∆1F , µm ), and η is a cocycle representing cyc1[u] ([u]). On
the other hand, it follows from the construction of the map cl1 that the element
1
cl1 ([u]) ∈ Hét
(F, µm ) is given as follows: Since i∗(1,0) ◦ i∗ = i∗(0,1) ◦ i∗ = 0, the map
i∗ extends to the map of complexes
ĩ∗ : G̃∗[u] (∆1F , µm ) → G∗ (∆∗F , µm ),
1
and cl1 ([u]) is the element of Hét
(F, µm ) corresponding to ĩ∗ (cyc1[u] ([u])) via the
∗
∼
quasi-isomorphism Gét (F, µm ) = G∗ (∆∗F , µm ) induced by the augmentation ∆∗F →
F . Tracing through this quasi-isomorphism, we see that cl1 ([u]) is the class of
i∗(1,0) (ψ) − i∗(0,1) (ψ), where ψ is any element of G1 (∆1F , µm ) with dψ = i∗ η. Since
we may take ψ = φ, and since i∗p (φ) = i∗p ({[f ]}) for all p ∈ U , it follows that cl1 ([u])
is the class of i∗(1,0) ({[f ]}) − i∗(0,1) ({[f ]}), which verifies (4.7).
1
Since we may take f to be the function f (t0 , t1 ) = t0 − 1−u
, we have
cl1 ([u]) =
1−
1
1−u
1
− 1−u
mod F ∗m = u
mod F ∗m .
26
THOMAS GEISSER AND MARC LEVINE
This result has the following consequence for the relation of the Bloch-Kato
conjecture and the cycle class map:
Lemma 4.10. Let F be a field, m an integer prime to the characteristic of F .
q
q
If ϑq,F : KqM (F )/m → Hét
(F, µ⊗q
m ) is surjective, then the cycle class map cl :
q
H q (F, Z/m(q)) → Hét (F, µ⊗q
m ) is surjective as well.
Proof. The Galois symbol for q = 1 is by definition the Kummer isomorphism
1
F ∗ /F ∗m ∼
(F, µm ), and the map ϑq,F is the multiplicative extension of ϑ1,F .
= Hét
Thus, ϑq,F is surjective if and only if the cup product map
q
1
Hét
(F, µm )⊗q → Hét
(F, µ⊗q
m )
is surjective. From Proposition 4.9 and Proposition 4.7, this implies the surjectivity
of clq .
Remark 4.11. One could also prove Lemma 4.10 by showing that the isomorphisms
KnM (F ) → H n (F, Z(n)) described in [20] or [26] are multiplicative. The map described in [26] uses a cubical version of the higher Chow groups, and the issue
of multiplicativity is not addressed in [20]; we have used the somewhat indirect
argument above to avoid checking this detail.
0
Using Proposition 4.9, one can show that cl1 : H 0 (F, Z/m(1)) → Hét
(F, µm ) is
an isomorphism as well. We are indebted to the referee for the argument.
Proposition 4.12. Let F be a field, and m an integer prime to char F . Then
the cycle class map cl1 : Z 1 (F, ∗) ⊗ Z/m → G∗ (F, µm ) is an isomorphism. In
0
particular, the map cl1 : H 0 (F, Z/m(1)) → Hét
(F, µm ) is an isomorphism.
Proof. We may assume that m is a prime power lν . As the map
cl1 : Z 1 (X, ∗) ⊗ Z/m → G∗ (X, µm )
is defined by a finite zigzag diagram of natural maps of complexes, cl1 is compatible
with inverse systems of schemes with flat affine transition maps, and induces a map
on the corresponding direct limit of complexes. Since both Z 1 (X, ∗) and G∗ (X, µm )
are compatible with such inverse systems, it suffices to prove the result for F finitely
generated over the prime field.
We now consider the inverse system of maps cl1 for m = ln , n = 1, 2, . . . . Since
the Godement resolution transforms surjective maps of sheaves to term-wise surjective maps of complexes, the complexes in the system of zigzag diagrams defining cl1
for m = ln , n = 1, 2, . . . , satisfy the Mittag-Leffler conditions term-wise. Therefore,
cl1 induces a map on the derived inverse limit
(4.9)
cl1 : R lim
Z 1 (F, ∗) ⊗ Z/ln → R lim
G∗ (F, µln ),
←
←
n
n
THE BLOCH-KATO CONJECTURE AND A THEOREM OF SUSLIN-VOEVODSKY
27
and we have the commutative diagram for each N
(4.10)
R lim
Z 1 (F, ∗) ⊗ Z/ln
←
/ Z 1 (F, ∗) ⊗ Z/lN
n
cl1
cl1
/ G∗ (F, µlN )
R lim
G∗ (F, µln )
←
n
where the horizontal arrows are the canonical maps.
n
Both Z 1 (F, ∗)⊗Z/ln and G∗ (F, µln ) have the same cohomology, namely, F ∗ /(F ∗ )l
in degree one, and µln (F ) in degree zero. Since F is finitely generated, both
R lim Z 1 (F, ∗) ⊗ Z/ln and R lim G∗ (F, µln ) have cohomology only in degree one,
←-
←-
with that cohomology being lim F ∗ /(F ∗ )l . By Proposition 4.9 and the commutan
←-
tive diagram (4.10), the map (4.9) is an isomorphism.
The commutativity of the diagram (4.10) identifies cl1 : Z 1 (F, ∗) ⊗ Z/lN →
∗
G (F, µlN ) with the map on the derived tensor product
cl1 ⊗id : [R lim
Z 1 (F, ∗) ⊗ Z/ln ] ⊗L Z/lN → [R lim
G∗ (F, µln )] ⊗L Z/lN .
←
←
n
n
As this latter map is an isomorphism, the proposition is proved.
4.13. The étale sheafification of ΓX (q). In this section, we prove Theorem 1.5,
and show that Theorem 1.1 and Theorem 1.6 are equivalent.
Let ΓX (q)ét denote the sheafification of ΓX (q) for the étale topology on X.
Sheafifying the cycle class map (3.20) for the étale topology on X as in §3.10 gives
the étale cycle class map
clqét : ΓX (q)ét ⊗L Z/m → µ⊗q
m
in D− (Sh(Xét )), natural in X. We have the natural map ΓX (q) → R∗ ΓX (q)ét ;
truncating gives us the natural map
ιX : ΓX (q) → τ≤q R∗ ΓX (q)ét .
Proof of Theorem 1.5. Let π : X → Spec k be the structure morphism. By the
rigidity result of [1, Lemma 11.1], the natural map
π ∗ Γk (q)ét ⊗L Z/m → ΓX (q)ét ⊗L Z/m
is a quasi-isomorphism, reducing us to the case of X = Spec k, with k separably
closed.
Suppose that char k = 0. By the Suslin-Voeovodsky theorem relating Suslin
homology and étale homology [24, Corollary 7.8] and Suslin’s comparison of Suslin
homology and motivic cohomology [23], we have
0
for p = 0
p
H (k, Z/m(q)) =
(k)
for p = 0.
µ⊗q
m
In case char k > 0, the same result follows by using de Jong’s resolution of singularities [11] to extend the results of [24] to positive characteristic (see e.g. [12] or
[7] for further details).
28
THOMAS GEISSER AND MARC LEVINE
0
By Proposition 4.12, the map cl1 : H 0 (k, Z/m(1)) → Hét
(k, µm ) is an isomorphism. By Proposition 4.7, we have the commutative diagram
H 0 (k, Z/m(1))⊗q
∪
(cl1 )⊗q
0
Hét
(k, µm )⊗q
/ H 0 (k, Z/m(q))
clq
∪
/ H 0 (k, µ⊗q )
m
ét
from which we see that the cup product H 0 (k, Z/m(1))⊗q → H 0 (k, Z/m(q)) is
injective. From the computation of H 0 (k, Z/m(q)) given above, this shows that
the product map H 0 (k, Z/m(1))⊗q → H 0 (k, Z/m(q)) is an isomorphism. Thus the
cycle class map induces a quasi-isomorphism
clq : Γk (q) ⊗L Z/m → µ⊗q
m (k),
for k separably closed, completing the proof.
We now show that Theorem 1.1 and Theorem 1.6 are equivalent.
Proof. By Theorem 1.5, we have the isomorphism
claét : ΓX (a)ét ⊗L Z/m → µ⊗a
m
in D− (Sh(Xét )) for all a ≥ 0. Applying R∗ and truncating gives the isomorphism
in D− (Sh(XZar ))
τ≤a R∗ claét : τ≤a R∗ ΓX (a)ét ⊗L Z/m → τ≤a R∗ µ⊗a
m .
By the adjunction property of R∗ , the cycle class map factors through τ≤a R∗ claét ,
giving the commutative diagram
ιX
/ τ≤a R∗ ΓX (a)ét ⊗L Z/m
ΓX (a) ⊗L Z/m
TTTT
TTTT
TTTT
τ≤a R.∗ cla
ét
TTT)
cla
τ≤a R∗ µ⊗a
m
with τ≤a R∗ claét an isomorphism. Thus cla is an isomorphism if and only if ιX is
an isomorphism, showing that Theorem 1.1 and Theorem 1.6 are equivalent.
5. The semi-local n-cube
We relate the Bloch-Kato conjecture to the Beilinson-Lichtenbaum conjectures
by taking motivic cohomology and étale cohomology of the semi-local n-cube, relative to its faces.
5.1. The semi-local n-cube. We write n for the affine space Spec k[t1 , . . . , tn ].
Let v be the set of points (1 , . . . , n ), j ∈ {0, 1}, of n , Rn the semi-local ring
ˆ n the semi-local scheme Spec Rn .
On ,v and i;.
ˆ n be the subscheme of ˆ n defined by the ideal (ti − ), ∈ {0, 1}. We
Let ˆ n by
define the set of subschemes Tn of ˆ ni;. | i = 1, . . . , n; ∈ {0, 1}}.
Tn := {
We order the elements of Tn by setting
ˆ s;1
ˆ
n,s := ˆ ns−n;0
n
for s ≤ n
for s > n,
THE BLOCH-KATO CONJECTURE AND A THEOREM OF SUSLIN-VOEVODSKY
29
and let Tns be the subset of Tn consisting of the first s subschemes. Let
ˆ n;0
Sn := Tn2n−1 = Tn − {
n }.
r
ˆ n,s of the form ˆ n,i ∩ ˆ n,s ,
For r < s, we let Tn,s
denote the set of subschemes of i = 1, . . . , r.
ˆ n,I for the intersection ∩s∈I ˆ n,s ; we call ˆ n,I a face of ˆ n.
As in §2.7, we write ˆ
ˆ
Using the coordinates tj , j ∈ I, in the standard order identifies n,I with n−|I| .
ˆ n,s of ˆ n with ˆ n−1 in this way, we have the identification
Identifying the face r
for r < s ≤ n or r + n < s,
Tn−1
r
Tn,s =
r−1
Tn−1
for 1 ≤ s − n ≤ r < s.
5.2. Relativization sequences. For 1 ≤ s ≤ 2n we have the fundamental relativization sequence
s−1
ˆ n,s ; Tn,s
ˆ n ; Tns , Z/m(q)) →
→ H p−1 (
, Z/m(q)) → H p (
s−1
ˆ n ; Tns−1 , Z/m(q)) → H p (
ˆ n,s ; Tn,s
H p (
, Z/m(q)) → .
We have similar sequences for étale cohomology. Taking s = 2n and identifying
ˆ n,2n with ˆ n−1 as described above gives the most important case
ˆ n−1 ; Tn−1 , Z/m(q)) → H p (
ˆ n ; Tn , Z/m(q)) →
→ H p−1 (
ˆ n ; Sn , Z/m(q)) → H p (
ˆ n−1 ; Tn−1 , Z/m(q)) → .
H p (
We will rely on the fundamental surjectivity property, proved in [8, Corollary
4.4]:
Lemma 5.3. For all n ≥ 1 and all q ≥ 0, the restriction map
ˆ n ; Sn , Z/m(q)) → H q (
ˆ n−1 ; Tn−1 , Z/m(q))
H q (
is surjective.
Remark 5.4. We have not been able to prove the analog of Lemma 5.3 for the semilocal scheme of the vertices in ∆n , which is why we need to introduce the n-cube
ˆ n . Lemma 5.3 is similar to the result [25, Corollary 9.7](revised version), and
plays the same crucial role in the argument.
ˆ n and its faces are similar to that of ∆n and its faces. In
The combinatorics of the next few sections, we discuss these combinatorics, leading to the splitting result
Proposition 5.7.
5.5. Relative complexes. Let C be a subcategory of Schk , and F : C op → C(Ab)
a functor. As in §3.1, if we have a k-scheme X with closed subschemes Y1 , . . . , Ym
such that all the inclusions YI → YJ , J ⊂ I ⊂ {1, . . . , n}, are in C, we form the
relative complex F (X; Y1 , . . . , Yn ), which is defined as the total complex of the
double complex
F (X) →
n
i=1
F (Yi ) → . . . →
|I|=j
F (YI ) → . . .
30
THOMAS GEISSER AND MARC LEVINE
with F i (YI ) in total degree i + |I|. The relative complexes F (X; Y1 , . . . , Yn ) are
natural in F , and have the functorialities described in §2.8. We have as well the
subcomplex
ker
F (X; Y1 , . . . , Yn )
n
:= ker F (X) →
F (Yi )
i=1
of F (X; Y1 , . . . , Yn ).
5.6. The category C(n). For 1 ≤ s ≤ n let
is : (t1 , . . . , tn−1 ) → (t1 , . . . , ts−1 , 0, ts , . . . , tn−1 )
js : (t1 , . . . , tn−1 ) → (t1 , . . . , ts−1 , 1, ts , . . . , tn−1 )
ˆ n . We have the identities
be the inclusion of the faces ts = 0 and ts = 1 into (5.1)
is+1 it
it is =
is it−1
is+1 jt
jt is =
is jt−1
s≥t
s < t;
s≥t
s < t;
js+1 jt
jt js =
js jt−1
s≥t
s < t.
Similarly, we define projection maps for 1 ≤ s ≤ n and 1 ≤ s < n, respectively,
ps : (t1 , . . . , tn ) → (t1 , . . . , ts−1 , ts+1 , . . . , tn )
qs : (t1 , . . . , tn ) → (t1 , . . . , ts−1 , 1 − (ts − 1)(ts+1 − 1), ts+2 , . . . , tn ).
The following identities hold


jt−1 ps
ps jt = id
(5.2)


jt ps−1
(5.3)


jt−1 qs
qs jt = js ps


jt qs−1
t>s
t=s
t < s;


it−1 ps
ps it = id


it ps−1
t>s
t=s
t < s,
t>s+1
t = s, s + 1
t < s;


it−1 qs
qs it = id


it qs−1
t>s+1
t = s, s + 1
t < s.
Let I be a subset of {1, . . . , n} (possibly empty), and s1 , . . . , sr elements of the
ˆ
ˆ n for the subscheme ∪r ˆ
complement of I. We write ∂Is1 ,... ,sr i=1 n,I∪{si } of n,I .
s1 ,... ,sr ˆ
s1 ,... ,sr ˆ
1,... ,2r ˆ
ˆ
We write ∂
n for ∂∅
n ; we also write ∂ n for ∂
n . We allow
ˆn = ˆ n are all normal
ˆ n,I . We note that the ∂ s1 ,... ,sr the case r = 0, i.e., ∂I I
ˆ n.
crossing subschemes of ˆ n is a face ˆ n . We
ˆ n,J of ˆ n which is contained in ∂ s1 ,... ,sr A face of ∂Is1 ,... ,sr I
ˆ n of the form
let C(n) be the subcategory of Schk with objects the subschemes of ˆ n . A morphism f : ∂ s1 ,... ,sr ˆ n → ∂ s1 ,... ,sr ˆ n is a map of k-schemes such
∂Is1 ,... ,sr I
I
ˆ n,J of ∂ s1 ,... ,sr ˆ n , there is a face ˆ n such
ˆ n,J of ∂ s1 ,... ,sr that, for each face I
I
ˆ n,J factors as
that the restriction of f to ˆ n,J −f→ ˆ n,J
J
i
s ,... ,sr
J
−−
→ ∂I 1
ˆ n ,
with fJ flat, and iJ the inclusion.
Proposition 5.7. Let F : C(n)op → C(Ab) be a functor. Then for all s < 2n:
THE BLOCH-KATO CONJECTURE AND A THEOREM OF SUSLIN-VOEVODSKY
1. The canonical maps
ˆ n , Tns )ker → F (
ˆ n , Tns );
F (
ˆ n , Tns )ker → F (∂ ˆ n , Tns )
F (∂ are quasi-isomorphisms.
2. The inclusions
ˆ n , Tns )ker ⊆ F (
ˆ n , Tns−1 )ker ;
F (
ˆ n , Tns )ker ⊆ F (∂ ˆ n , Tns−1 )ker
F (∂ 31
are split; the splittings are natural in F .
Proof. We can assume that the proposition holds for functors C(n − 1)op → C(Ab)
ˆ n ; the proof
and proceed by induction on s. We will prove the proposition for ∂ ˆ
ˆ
ˆ
for n is exactly the same, replacing ∂ n with n , and is left to the reader.
By contravariant functoriality, there are maps
ˆ n ) → F (
ˆ n−1,s )
i∗s , js∗ : F (∂ ˆ n−1,s ) → F (∂ ˆ n ).
p∗s , qs∗ : F (
For s < 2n, consider the following commutative diagram of complexes:
ˆ n , Tns )ker
F (∂ (5.4)
incl
/ F (∂ ˆ n , Tns−1 )ker
α
β
ˆ n , Tns )
F (∂ φ
s−1 ker
/ F (
ˆ n,s , Tn,s
)
γ
s−1
/ F (
ˆ n,s , Tn,s
),
/ F (∂ ˆ n , Tns−1 )
where the vertical arrows are the natural inclusions, and φ is the restriction map.
ˆ n , Tns )ker = ker(φ).
By definition, F (∂ The bottom row of (5.4) defines a distinguished triangle via the isomorphism
(2.10). If we can show that the map φ is naturally split surjective, then (2) follows,
and the top row of (5.4) defines a distinguished triangle as well. Since β and γ
are quasi-isomophisms by induction, α will be a quasi-isomorphism as well, proving
(1).
We consider the following diagram, where ιs = js∗ for s ≤ n and ιs = i∗s−n for
s > n, and where β̃ and γ̃ are the evident inclusions:
0
/ F (∂ ˆ n , T s−1 )ker
β̃
n
0
Q
t≤s−1 ιt
/
γ̃
/ F (
ˆ n,s )
t≤s−1
ˆ n,t )
F (
⊕ιs
ιs
φ
s−1 ker
/ F (
ˆ n,s , Tn,s
)
/ F (∂ ˆ n)
Q
t≤s−1 ιt
/
ˆ
F
t≤s−1 (n,t,s ).
ˆ n , Tns−1 )ker
The rows are the degree-wise exact sequences defining the complexes F (∂ s−1 ker
ˆ
and F (∂ n , Tn ) , respectively.
∗
ˆ n,s ) → F (∂ ˆ n ) be the map p∗s in case s ≤ n, or qn−s
Let ρs : F (
in case
s−1 ker
ˆ n,s , Tn,s
n < s < 2n. Let x be an element of F (
) . Then x can be viewed as an
ˆ n,s ) mapping to zero under ιt for t < s. Map x to x̄ in F (∂ ˆ n ) by
element of F (
taking x̄ = ρs (x).
We note that ιs ◦ ρs = id, by (5.2) for s ≤ n or (5.3) for n < s < 2n. Thus,
to show that ρs defines our desired splitting, we need only show that x̄ lies in the
kernel of ιt for t < s. But this follows from (5.2) for t < s ≤ n:
ιt x̄ = jt∗ p∗s x = p∗s−1 jt∗ x = 0.
32
THOMAS GEISSER AND MARC LEVINE
For s > t > n this follows from (5.3):
∗
∗
x = qs−n−1
i∗t−n x = 0,
ιt x̄ = i∗t−n qs−n
and for t ≤ n < s,

∗
∗

qs−n jt−1 x = 0 t > s − n + 1
∗
ιt x̄ = jt∗ qs−n
x = p∗s−n jt∗ x = 0
t = s − n, s − n + 1

 ∗
qs−n−1 jt∗ x = 0 t < s − n.
ˆn
6. The Beilinson-Lichtenbaum conjectures and cycle maps for In the main result of this section, Proposition 6.5, we show how the surjectivity
of certain relative cycle class maps for semi-local n-cubes implies a part of the
Beilinson-Lichtenbaum conjectures for fields.
We have the following version of a part of the Beilinson-Lichtenbaum conjectures
[16]:
Conjecture 6.1 (BLa ). Let m be an integer prime to the characteristic of k. The
cycle class map
p
cla : H p (F, Z/m(a)) → Hét
(F, µ⊗a
m )
is an isomorphism for all p ≤ a, and for all finitely generated field extensions F of
k.
Write A for Ank , and let A(b) denote the set of codimension b points of A which
ˆ n . The standard constructions of Bloch-Ogus [4] give the augmented
are not in Gersten complex for motivic cohomology
ˆ n , Z/m(q)) →
(6.1) 0 → H p (A, Z/m(q)) −
→ H p (
.
→ ... →
x∈A
x∈A(1)
H p−1 (k(x), Z/m(q − 1))
H p−r (k(x), Z/m(q − r)) → . . .
(r)
and for étale cohomology
.
p
p ˆ
(6.2) 0 → Hét
(A, µ⊗q
→ Hét
(n , µ⊗q
m )−
m )→
x∈A(1)
→ ... →
p−1
Hét
(k(x), µ⊗q−1
)
m
x∈A(r)
p−r
Hét
(k(x), µ⊗q−r
) → ...
m
p
To fix the notation, we take H p (A, Z/m(q)) and Hét
(A, µ⊗q
m ) in degree −1.
Lemma 6.2. The complexes (6.1) and (6.2) are exact.
Proof. We discuss the case of motivic cohomology; the argument for étale cohomology is exactly the same.
THE BLOCH-KATO CONJECTURE AND A THEOREM OF SUSLIN-VOEVODSKY
33
For a scheme X, we let X (r) denote the set of codimension r points of X. We
have the standard Gersten complex for A:
(6.3) 0 → H p (A, Z/m(q)) → H p (k(A), Z/m(q)) →
H p−1 (k(x), Z/m(q − 1))
(1)
x∈A
p−r
→ ... →
H
(k(x), Z/m(q − r)) → . . . ,
x∈A(r)
ˆ n:
and for (6.4)
ˆ n , Z/m(q)) → H p (k(
ˆ n ), Z/m(q)) →
0 → H p (
→ ... →
(1)
H p−1 (k(x), Z/m(q − 1))
x∈ ˆ n
H p−r (k(x), Z/m(q − r)) → . . . .
(r)
x∈ ˆ n
Sherman [21] has considered the analog of (6.3), where one replaces motivic cohomology with K-cohomology H ∗ (−, K∗ ), and shows that the resulting complex
is exact. The same argument works for any Bloch-Ogus twisted duality theory, in
particular, the complex (6.3) is exact. Similarly, the “classical” Gersten’s lemma
[4, Theorem 4.2] for a Bloch-Ogus cohomology theory shows that (6.4) is exact.
We define the map of complexes ψ : (6.3) → (6.4) by the restriction map
ˆ n , Z/m(q)),
ψ −1 : H p (A, Z/m(q)) → H p (
in degree −1 and the evident projection
ψr :
H p−r (k(x), Z/m(q − r)) →
H p−r (k(x), Z/m(q − r))
(r)
ˆ (r)
x∈A
x∈
in degree r ≥ 0. It is easy to check that the complex (6.1) is homotopy equivalent
to cone(ψ)[−1], which shows that (6.1) is exact.
ˆ n.
We write 1 for the point (1, . . . , 1) of Lemma 6.3. Suppose BLa is true for 0 ≤ a < q. Then the cycle class map
ˆ n ; 1, Z/m(q)) → H p (
ˆ n ; 1, µ⊗q
clq : H p (
m )
ét
is an isomorphism for all p ≤ q and all n ≥ 1.
Proof. By Lemma 6.2, the Gersten complexes (6.1) and (6.2) are exact.
ˆ n, j : ˆ n → A the
Let π : A → Spec k be the structure morphism, and i : 1 → inclusions. Since
p
H p (k, Z/m(q)) −→ Hét
(A, Z/m(q));
π∗
p
p
⊗q
Hét
(k, µ⊗q
m ) −→ Hét (A, µm )
π∗
are isomorphisms, and π ◦ j ◦ i = id, we may split off the first term in both (6.1)
and (6.2), giving the exact Gersten complexes
ˆ n ; 1, Z/m(q)) →
(6.5) 0 → H p (
H p−1 (k(x), Z/m(q − 1)) → . . .
x∈A(1)
→
H p−r (k(x), Z/m(q − r)) → . . .
x∈A(r)
34
THOMAS GEISSER AND MARC LEVINE
and
p ˆ
(6.6) 0 → Hét
(n ; 1, µ⊗q
m )→
x∈A(1)
p−1
Hét
(k(x), µ⊗q−1
) → ...
m
→
x∈A
p−r
Hét
(k(x), µ⊗q−r
) → ...
m
(r)
The compatibility of the cycle class maps with localization (Proposition 4.5(2))
implies that the various cycle class maps give a map of complexes cl∗ : (6.5) → (6.6).
This together with our hypothesis proves the lemma.
Lemma 6.4. Suppose that BLa is true for 0 ≤ a < q. Then the cycle class map
ˆ n ; Sn , Z/m(q)) → H p (
ˆ n ; Sn , µ⊗q
clq : H p (
m )
ét
is an isomorphism for all p ≤ q.
Proof. We prove more generally that the cycle class map
ˆ n ; Tns , Z/m(q)) → H p (
ˆ n ; Tns , µ⊗q
clq : H p (
m )
ét
is an isomorphism for all p ≤ q and for 1 ≤ s < 2n. We proceed by induction on s
and n.
By Proposition 4.5(1), we have the commutative diagram
ˆ n ; Tns , Z/m(q))
→ H p (
clq
/ H p (
ˆ n ; Tns−1 , Z/m(q))
s−1
/ H p (
ˆ n,s ; Tn,s
, Z/m(q)) →
clq
p ˆ
→ Hét
(n ; Tns , µ⊗q
m )
clq
s−1
/ H p (
ˆ n,s ; Tn,s
, µ⊗q
m ) →,
ét
/ H p (
ˆ n ; Tns−1 , µ⊗q
m )
ét
where the rows are the respective relativization sequences. Induction reduces us to
the case s = 1.
We have the relativization sequence
ˆ n ; Tn1 , 1, Z/m(q)) → H p (
ˆ n ; Tn1 , Z/m(q)) → H p (1; 1, Z/m(q)) →;
→ H p (
since Z q (1; 1, ∗) is the cone on the identity map, we thus have the isomorphism
ˆ n ; Tn1 , 1, Z/m(q)) ∼
ˆ n ; Tn1 , Z/m(q)),
H p (
= H p (
and similarly for relative étale cohomology. Comparing the relativization sequences
for motivic cohomology and étale cohomology using Proposition 4.5(1) gives the
commutative diagram
ˆ n ; Tn1 , 1, Z/m(q))
→ H p (
clq
p ˆ
→ Hét
(n ; Tn1 , 1, µ⊗q
m )
/ H p (
ˆ n ; 1, Z/m(q))
/ H p (
ˆ n−1 ; 1, Z/m(q)) →
clq
/ H p (
ˆ n ; 1, µ⊗q
m )
ét
clq
/ H p (
ˆ n−1 ; 1, µ⊗q
m )→.
ét
Using Lemma 6.3, this shows that the cycle class map
ˆ n ; Tn1 , 1, Z/m(q)) → H p (
ˆ n ; Tn1 , 1, µ⊗q
clq : H p (
m )
ét
is an isomorphism in the desired range.
THE BLOCH-KATO CONJECTURE AND A THEOREM OF SUSLIN-VOEVODSKY
35
Proposition 6.5. Suppose that BLa is true for 0 ≤ a < q. Suppose further that,
for all finitely generated field extensions F of k, the cycle class maps
ˆ n ×k F ; Tn ×k F, Z/m(q)) → H q (
ˆ n ×k F ; Tn ×k F, µ⊗q
clq : H q (
m )
ét
are surjective for all n ≥ 0. Then BLq is true.
Proof. We first reduce BLq to showing
(6.7)
For all finitely generated field extensions F of k, the cycle class maps
ˆ n ×k F ; Tn ×k F, Z/m(q)) → H p (
ˆ n ×k F ; Tn ×k F, µ⊗q
clq : H p (
m )
ét
are isomorphisms for all n ≥ 1 and all p ≤ q.
Indeed, suppose that (6.7) is true. If k is a finitely generated field extension of
k, (6.7) remains true with k replacing k. Thus, it suffices to prove that
p
(Spec k, µ⊗q
clq : H p (Spec k, Z/m(q)) → Hét
m )
is an isomorphism for p ≤ q.
We have the commutative diagram relating the motivic and étale relativization
sequences
/ H p (
ˆ 1 ; T1 , Z (q)) / H p (
ˆ 1 ; S1 , Z (q)) / H p (k, Z (q)) / H p+1 (
ˆ 1 ; T1 , Z (q)) →
m
m
m
m
clq
clq
/ H p (
ˆ 1 ; T1 , µ⊗q
m )
ét
clq
/ H p (
ˆ 1 ; S1 , µ⊗q
m )
ét
/ H p (k, µ⊗q
m )
ét
clq
/ H p+1 (
ˆ 1 ; T1 , µ⊗q
m )→
ét
With this, (6.7) together with Lemma 6.4 and (for p = q) Lemma 5.3 implies that
clq is an isomorphism in the desired range.
To prove (6.7), we may replace F with k. We consider commutative diagram of
relativization sequences
ˆ n+1 ; Tn+1 , Z (q))
→ H q (
m
clq
/ H q (
ˆ n+1 ; Sn+1 , Z (q))
m
clq
q ˆ
→ Hét
(n+1 ; Tn+1 , µ⊗q
m )
/ H q (
ˆ n+1 ; Sn+1 , µ⊗q
m )
ét
/ H q (
ˆ n ; Tn , Z (q))
m
/0
clq
/ H q (
ˆ n ; Tn , µ⊗q
m )
ét
/
where the surjectivity for motivic cohomology is given by Lemma 5.3. The map
ˆ n ; Sn , Z/m(q)) → H q (
ˆ n ; Sn , µ⊗q
clq : H q (
m )
ét
is an isomorphism by Lemma 6.4, hence surjectivity of
ˆ n+1 ; Tn+1 , Z/m(q)) → H q (
ˆ n+1 ; Tn+1 , µ⊗q
clq : H q (
m )
ét
ˆ n ; Tn , Z/m(q)) → H q (
ˆ n ; Tn , µ⊗q
implies injectivity of clq : H q (
m ).
ét
We now proceed by descending induction on the cohomology degree. Fix an
ˆ n ; Tn , Z/m(q)) → H a (
ˆ n ; Tn , µ⊗q
integer p < q, and suppose that clq : H a (
m ) is an
ét
isomorphism for all a with p + 1 ≤ a ≤ q and for all n. We have the commutative
36
THOMAS GEISSER AND MARC LEVINE
diagram whose columns are the respective relativization sequences
ˆ n+1 ; Tn+1 , Z/m(q))
H p (
clq
/ H p (
ˆ n+1 ; Tn+1 , µ⊗q
m )
ét
ˆ n+1 ; Sn+1 , Z/m(q))
H p (
clq
/ H p (
ˆ n+1 ; Sn+1 , µ⊗q
m )
ét
ˆ n ; Tn , Z/m(q))
H p (
clq
/ H p (
ˆ n ; Tn , µ⊗q
m )
ét
ˆ n+1 ; Tn+1 , Z/m(q))
H p+1 (
clq
/ H p+1 (
ˆ n+1 ; Tn+1 , µ⊗q
m )
ét
ˆ n+1 ; Sn+1 , Z/m(q))
(
clq
/ H p+1 (
ˆ n+1 ; Sn+1 , µ⊗q
m )
ét
H
p+1
As we have the isomorphism (Lemma 6.4)
a ˆ
ˆ n+1 ; Sn+1 , Z/m(q)) → Hét
clq : H a (
(n+1 ; Sn+1 , µ⊗q
m )
ˆ n ; Tn , Z/m(q)) → H p (
ˆ n ; Tn , µ⊗q
for all a ≤ q, the map clq : H p (
m ) is surjective;
ét
as n is arbitrary, the map
ˆ n+1 ; Tn+1 , Z/m(q)) → H p (
ˆ n+1 ; Tn+1 , µ⊗q
clq : H p (
m )
ét
ˆ n ; Tn , Z/m(q)) → H p (
ˆ n ; Tn , µ⊗q
is surjective as well, hence cl : H (
m ) is injective.
ét
q
p
7. The Bloch-Kato conjecture and surjectivity
We now complete the discussion, by showing how the Bloch-Kato conjecture for
fields implies the surjectivity condition of Proposition 6.5.
Proposition 7.1. Let X = Spec R be a semi-local k-scheme, where R is a localp
ization of a k-algebra of finite type. Then, for each element η of Hét
(X, µ⊗q
m ), there
is a k-morphism i : X → Y , with Y a smooth finite-type semi-local k-scheme, and
p
∗
an element τ of Hét
(Y, µ⊗q
m ), such that η = i τ .
Proof. Take a closed embedding of X into the semi-localization A of an affine space
ANk , and let Xh be the henselization of A along X. Let ih : X → Xh be the inclusion
and let F be a torsion étale sheaf on Xh . It follows from [6, Theorem 1] (see also
p
p
[22]) that i∗h : Hét
(Xh , F) → Hét
(X, i∗h F) is an isomorphism. Since X and A are
affine, Xh is a filtered projective limit of smooth semi-local k-schemes of finite type,
p
with flat affine transition maps. Since Hét
(−, µ⊗q
m ) maps filtered projective limits
of this type to filtered inductive limits, the result follows.
Proposition 7.2. Suppose that, for all fields F finitely generated over k, the cycle
q
class map clq : H q (F, Z/m(q)) → Hét
(F, µ⊗q
m ) is a surjection and the cycle class
q−1
q−1
q−1
map cl
: H
(F, Z/m(q − 1)) → Hét
(F, µ⊗q−1
) is an injection. Then the
m
cycle class map
ˆ n ×k F ; Tn ×k F, Z/m(q)) → H q (
ˆ n ×k F ; Tn ×k F, µ⊗q
clq : H q (
m )
ét
is surjective for all fields F finitely generated over k and for all n.
THE BLOCH-KATO CONJECTURE AND A THEOREM OF SUSLIN-VOEVODSKY
37
Proof. As in the proof of Proposition 6.5, it suffices to prove the result for F = k.
By a limit argument, similar to the argument of §3.4, we may assume that k is
perfect.
Let R be the semi-local ring of finitely many smooth points on a algebraic variety
X over k. By Bloch-Ogus theory [4, Theorem 4.2], we have the exact Gersten
complex
(7.1) 0 → H p (R, Z/m(q)) → H p (k(X), Z/m(q)) → . . .
→
H p−a (k(x), Z/m(q − a)) → . . .
x∈(Spec R)(a)
and a similar exact Gersten complex for étale cohomology
p
p
⊗q
(7.2) 0 → Hét
(R, µ⊗q
m ) → Hét (k(X), µm ) → . . .
→
p−a
Hét
(k(x), µ⊗q−a
) → ...
m
x∈(Spec R)(a)
By the compatibility of the cycle class maps with localization (Proposition 4.5(2)),
the maps clq−a define the map of complexes cl∗ : (7.1) → (7.2).
Taking p = q and using our assumption, this shows that the cycle class map clq :
q
q
(R, µ⊗q
H (R, Z/m(q)) → Hét
m ) is surjective. By Proposition 7.1, and the naturality
ˆ n+1 , Z/m(q)) →
of the cycle class map (Proposition 4.2(5)), the map clq : H q (∂ q
⊗q
ˆ
Hét (∂ n+1 , µm ) is surjective for all n. We have the commutative diagram
(7.3)
ˆ n+1 , Z/m(q))
H q (∂ clq
s
q
ˆ
H (∂ n+1 ; Sn+1 , Z/m(q))
i∗
ˆ n ; Tn , Z/m(q))
H q (
/ H q (∂ ˆ n+1 , µ⊗q
m )
ét
s
cl
q
clq
/ H q (∂ ˆ n+1 ; Sn+1 , µ⊗q
m )
ét
i∗
/ H q (
ˆ n ; Tn , µ⊗q
m )
ét
where s is the natural splitting given by Lemma 5.7, and i is the inclusion of the
face tn+1 = 0. Thus, the map
(7.4)
ˆ n+1 ; Sn+1 , Z/m(q)) → H q (∂ ˆ n+1 ; Sn+1 , µ⊗q
clq : H q (∂ m )
ét
is surjective.
We recall that étale cohomology satisfies excision for unions of closed subschemes.
Indeed, suppose Z = Z1 ∪ Z2 , with Zi closed in Z. The Mayer-Vietoris property
(see e.g. the proof of Lemma 3.6) implies we have the distinguished triangle
∗
⊗q
∗
⊗q
∗
⊗q
G∗ (Z, µ⊗q
m ) → G (Z1 , µm ) ⊕ G (Z2 , µm ) → G (Z1 ∩ Z2 , µm ) →
from which it follows that the natural map
∗
⊗q
∗
⊗q
∗
⊗q
cone(G∗ (Z, µ⊗q
m ) → G (Z2 , µm )) → cone(G (Z1 , µm ) → G (Z1 ∩ Z2 , µm ))
is a quasi-isomorphism. Thus, the restriction morphism
∗
∗
⊗q
i∗Z1 : Hét
(Z, Z2 , µ⊗q
m ) → Hét (Z1 , Z1 ∩ Z2 , µm )
is an isomorphism.
38
THOMAS GEISSER AND MARC LEVINE
q
q ˆ
⊗q
ˆ n+1 ; Sn+1 , µ⊗q
In particular, the map i∗ : Hét
(∂ m ) → Hét (n ; Tn , µm ) is an
isomorphism. The surjectivity of (7.4) and the commutativity of (7.3) thus prove
the proposition.
We conclude this sequence of results on surjectivity with the following elementary
but useful result:
q
Lemma 7.3. The surjectivity of clqF : H q (F, Z/m(q)) → Hét
(F, µ⊗q
m ) for all fields
a
F finitely generated over k implies the surjectivity of clF : H a (F, Z/m(a)) →
a
Hét
(F, µ⊗a
m ) for all fields F finitely generated over k and for all a with 0 ≤ a ≤ q.
Proof. We proceed by downward induction on a. Let R be the local ring F [X](X) ,
a+1
a
∗
a
⊗a
let η be in Hét
(F, µ⊗a
m ), and assume that clF (X) is surjective. Let p : Hét (F, µm ) →
a
Hét
(F (X), µ⊗a
m ) be the map induced by the inclusion F → F (X), and let ω =
1
∗
cl (X) ∪ p η. By Proposition 4.5, we have the commutative diagram
H a+1 (F (X), Z/m(a + 1))
∂
cla+1
F (X)
a+1
Hét
(F (X), µ⊗a+1
)
m
/ H a (F, Z/m(a))
cla
F
∂
/ H a (F, µ⊗a ),
m
ét
where the maps ∂ are the boundary maps in the localization/Gysin sequence for
the open complement Spec F (X) of Spec F in Spec F [X](X) . For a = 0, we have
canonical isomorphisms
H 0 (F, Z/m(0)) ∼
= Z/m ∼
= H 0 (F, Z/m),
ét
and
cl0F
is the identity. In addition, the map
∂ : H 1 (F (X), Z(1)) ∼
= F (X)× → H 0 (F, Z)
is just the classical divisor map, hence
∂(cl1 (X)) = cl0 (∂(X)) = cl0 (1) = 1.
∗
(F, µ⊗∗
Since the boundary map is a Hét
m )-module homomorphism, this gives the
identity η = ∂(ω). By assumption, there is an element z ∈ H a+1 (F (X), Z/m(a+1))
with cla+1 (z) = ω, giving claF (∂z) = ∂(cla+1
F (X) (z)) = ∂(ω) = η.
7.4. The main theorem. We can now give the proof of our main result Theorem 1.1. By Lemma 4.10, it suffices to prove
q
Theorem 7.5. Suppose that the maps clq : H q (F, Z/m(q)) → Hét
(F, µ⊗q
m ) are
surjective for all fields F finitely generated over k. The the cycle class map (1.2)
is an isomorphism for all essentially smooth X over k and all a with 0 ≤ a ≤ q.
Let R be the local ring of a smooth point on a k-variety X of finite type. To
prove Theorem 7.5, it suffices to show that the map
p
cla : H p (R, Z/m(a)) → Hét
(R, µ⊗a
m )
is an isomorphism for all p ≤ a ≤ q. We first reduce to the case of R a field which
is finitely generated over k.
As in the proof of Proposition 7.2, we have the exact (augmented) Gersten
complex for motivic cohomology (7.1), the exact (augmented) Gersten complex
THE BLOCH-KATO CONJECTURE AND A THEOREM OF SUSLIN-VOEVODSKY
39
for étale cohomology (7.2), and the cycle class maps give the map of complexes
cl∗ : (7.1) → (7.2). As this map in degree a ≥ 0 is
p−a
clq−a :
H p−a (k(x), Z/m(q − a)) → Hét
(k(x), µ⊗q−a
),
m
x∈Spec R(a)
the result for fields which are finitely generated over k yields the general case.
We now handle the case of fields which are finitely generated over k. We proceed
by induction on q, the case q = 0 being trivially satisfied.
By Lemma 7.3, the hypothesis in Theorem 7.5 implies that
a
cla : H a (F, Z/m(a)) → Hét
(F, µ⊗a
m )
is surjective for all fields F finitely generated over k and for all a with 0 ≤ a ≤ q.
Using our induction hypothesis, this implies that
(7.5)
p
cla : H p (F, Z/m(a)) → Hét
(F, µ⊗a
m )
is an isomorphism for all fields F finitely generated over k, and for p ≤ a < q.
Proposition 6.5 and Proposition 7.2 together with the isomorphisms (7.5) complete
the proof.
Remark 7.6. The idea of using Gabber’s rigidity theorem to pass from a version of
the Bloch-Kato conjecture for singular schemes to the usual version can be traced
back to R. Hoobler [10], who used an argument of this type to extend the MerkurjevSuslin theorem (which is the Bloch-Kato conjecture for weight two) to arbitrary
semi-local rings.
7.7. We conclude by proving Corollary 1.2. Let Hb (Z/m(q)) be the Zariski sheaf
b
of motivic cohomology on X, and Hét
(µ⊗q
m ) the Zariski sheaf of étale cohomology
on X. We have the spectral sequences
E2a,b = H a (XZar , Hb (Z/m(q))) =⇒ H a+b (X, Z/m(q))
a,b
a+b
b
⊗q
E2,ét
= H a (XZar , Hét
(µ⊗q
m )) =⇒ Hét (X, µm ).
∗∗
The cycle class map gives a map of spectral sequences clq : E ∗∗ → Eét
. If the
q
b
hypotheses of Corollary 1.2 are satisfied, then, by Theorem 1.1, cl : H (Z/m(q)) →
a,b
a,b
q
b
Hét
(µ⊗q
→ E2,ét
is an isomorphism
m ) is an isomorphism for b ≤ q, hence cl : E2
a,b
a,b
a,b
for b ≤ q. Since E2a,b = E2,ét
= 0 for a < 0, this implies that clq : E∞
→ E∞,ét
is
b
an isomorphism for a + b ≤ q, and for a + b = q + 1, b ≤ q. Since H (Z/m(q)) = 0
n
(X, µ⊗q
for b > q, this shows that clq : H n (X, Z/m(q)) → Hét
m ) is an isomorphism
for n ≤ q and an injection for n = q + 1, completing the proof.
8. Appendix–Products
The cycle complexes have a natural external product which can be constructed
along the lines described in [1]. There is a gap in the construction given in loc. cit.,
in that a part of the construction (Lemma (5.0)) relies on the incorrect proof of the
“moving lemma” [1, Theorem (3.3)]. Although the results of [2] give a proof of [1,
Theorem (3.3)], it is not clear that the argument given in [2] can be used to prove
Lemma (5.0). Therefore, in this Appendix, we give a construction of the product
on the cycle complexes.
Let X be a scheme, essentially of finite type over k. Let z q (X, p, p ) be the free
abelian group on the irreducible codimension q closed subsets W of X × ∆p × ∆p
40
THOMAS GEISSER AND MARC LEVINE
such that each irreducible component of W ∩ X × A × B has codimension q on
X × A × B for each face A of ∆p and B of ∆p . The z q (X, p, p ) form in the evident
q
way a bisimplicial abelian group; let z (X, ∗, ∗) be the associated double complex,
and let Tot z q (X, ∗, ∗) be the total complex of z q (X, ∗, ∗).
We give the product set [p] × [q] the product partial order,
(a, b) ≤ (a , b ) ⇐⇒ a ≤ a and b ≤ b ,
and identify [p] × [q] with the vertices of ∆p × ∆q in the evident way. For an orderpreserving map g : [n] → [p] × [q], the unique affine-linear extension of g gives the
map
∆(g) : ∆n → ∆p × ∆q .
A face of ∆p × ∆q is a subscheme of the form ∆(g)(∆n ) for some injective g.
Let z q (X, p, p )T be the subgroup of z q (X, p, p ) generated by the irreducible W ⊂
X × ∆p × ∆p such that, for each face A of ∆p × ∆p , each irreducible component
of W ∩ X × A has codimension q on X × A. The z q (X, p, p )T form a sub-double
complex z q (X, ∗, ∗)T of z q (X, ∗, ∗).
If g = (g1 , g2 ) : [p+q] → [p]×[q] is injective, g determines a p-q-shuffle by sending
i ∈ {1, . . . , p + q} to g1 (i) if g1 (i − 1) < g1 (i), and to g2 (i) + p if g2 (i − 1) < g2 (i).
Taking the sign of this permutation defines the sign sgn(g). Let
be the map
g
Tp,q : z r (X, p, q)T → z r (X, p + q)
sgn(g)∆(g)∗ , where the sum is over g as above. The map
T :=
Tp,q : Tot z r (X, ∗, ∗)T → z r (X, ∗),
p,q
is a well-defined map of complexes.
Lemma 8.1. The inclusion i : Tot z q (X, ∗, ∗)T → Tot z q (X, ∗, ∗) is a quasi-isomorphism.
Proof. We have the standard E 1 spectral sequence (of homological type) for each
of the double complexes z q (X, ∗, ∗)T , z q (X, ∗, ∗),
1
Ea,b
= Hb (z q (X, a, ∗)) =⇒ Ha+b (Tot z q (X, ∗, ∗)),
T
1
Ea,b
= Hb (z q (X, a, ∗)T ) =⇒ Ha+b (Tot z q (X, ∗, ∗)T );
the inclusion i induces the map of spectral sequences T E → E. It thus suffices to
show that the inclusion ia : z q (X, a, ∗)T → z q (X, a, ∗) is a quasi-isomorphism for
each a. The rest of the argument is similar to the proof of [1, Theorem 2.1]; we
now proceed to give the necessary modifications.
The complex z q (X, a, ∗) is evidently a subcomplex of z q (X × ∆a , ∗). It thus
suffices to show that the two inclusions z q (X, a, ∗) → z q (X × ∆a , ∗), z q (X, a, ∗)T →
z q (X × ∆a , ∗), are quasi-isomorphisms.
The coordinates t1 , . . . , ta on ∆a give an isomorphism of ∆a with Aa , and thereby
define an action of the group scheme (Aa , +, 0) on ∆a by translation. Let t be the
generic point of Aa , K the field k(t), and φt : ∆aK → ∆aK the translation by t.
Let Φ : ∆aK × ∆1 → ∆aK be the map Φ(x, (u0 , u1 )) = φu1 t (x) = x + u1 t. We let
π : X × ∆aK → X × ∆a be the projection, and i0 , i1 : X × ∆aK → X × ∆aK × ∆1 the
inclusions i0 (y) = (y, (1, 0)), i1 (y) = (y, (0, 1)). Note that Φ ◦ i0 = id, Φ ◦ i1 = φt .
THE BLOCH-KATO CONJECTURE AND A THEOREM OF SUSLIN-VOEVODSKY
41
If F is a face of ∆a × ∆p , then the orbit Aa · F is of the form ∆a × F for some
face F of ∆p . The argument of [1, Lemma 2.2] shows that the composition
Φ∗ ◦ π ∗ : z q (X × ∆a , ∗) → z q (X × ∆a × ∆1K , ∗)
has image in the subcomplex z q (X × ∆aK , 1, ∗)T of z q (X × ∆a × ∆1K , ∗), and that
the composition
i∗
Φ∗ ◦π ∗
1
z q (X × ∆a , ∗) −−−−→ z q (X × ∆aK , 1, ∗)T −→
z q (X × ∆aK , ∗),
which is just φ∗t ◦ π ∗ , has image in z q (XK , a, ∗)T .
We have the map T 1 : z q (X ×∆aK , 1, ∗)T → z q (X ×∆aK , ∗+1) induced by T . The
argument of [1, Lemma 2.2] shows in addition that T 1 ◦ Φ∗ ◦ π ∗ gives a homotopy
between the compositions
π ∗ , φ∗t ◦ π ∗ : z q (X × ∆a , ∗) → z q (X × ∆aK , ∗),
the compositions
π ∗ , φ∗t ◦ π ∗ : z q (X, a, ∗) → z q (XK , a, ∗),
and the compositions
π ∗ , φ∗t ◦ π ∗ : z q (X, a, ∗)T → z q (XK , a, ∗)T .
This implies that the maps
π̄ ∗ : z q (X × ∆a , ∗)/z q (X, a, ∗) → z q (XK × ∆a , ∗)/z q (XK , a, ∗)
π̄ ∗ : z q (X × ∆a , ∗)/z q (X, a, ∗)T → z q (XK × ∆a , ∗)/z q (XK , a, ∗)T
induce zero on homology. Since Spec K is a filtered inverse limit of finite type
k-schemes U with U (k) = ∅, and since the functors
U → z q (X × U × ∆a , ∗),
U → z q (X × U, a, ∗)
U → z q (X × U, a, ∗)T
transform filtered inverse limits of schemes with flat transition maps to direct limits
of complexes, it follows that the maps π̄ ∗ are injective on homology: for each such U ,
the choice of a k-point of U gives a left inverse to the maps induced by the projection
X × U → X. Thus z q (X × ∆a , ∗)/z q (X, a, ∗) and z q (X × ∆a , ∗)/z q (X, a, ∗)T are
acyclic, as desired.
Now let X and Y be schemes, essentially of finite type over k. Sending W ∈
z q (X × ∆p ) and W ∈ z q (Y × ∆p ) to the “product” cycle W × W ∈ z q (X ×k Y ×
∆p × ∆p ) defines the map of complexes
: zq (X, ∗) ⊗ zq (Y, ∗) → Tot zq+q (X ×k Y, ∗, ∗).
We have the diagram
i
(8.1) z q (X, ∗) ⊗ z q (Y, ∗) −→ Tot z q+q (X ×k Y, ∗, ∗) ←
− Tot z q+q (X ×k Y, ∗, ∗)T
T
−
→ z q+q (X ×k Y, ∗).
Via (8.1) and Lemma 8.1, the composition T ◦ i−1 ◦ defines the external product
map in D− (Ab)
(8.2)
∪X,Y : z q (X, ∗) ⊗L z q (Y, ∗) → z q+q (X ×k Y, ∗).
42
THOMAS GEISSER AND MARC LEVINE
The associativity and commutativity of ∪∗,∗ in D− (Ab) follow easily from wellknown associativity and commutativity properties of the triangulation T (see e.g.
[17, Chap. 3]); we leave the details to the reader.
References
[1] Bloch, S. Algebraic cycles and higher K-theory, Adv. in Math. 61 (1986), 267-304
[2] Bloch, S. The moving lemma for higher Chow groups J. Algebraic Geom. 3 (1994), no. 3,
537-568.
[3] Bloch, S., Kato, K. p-adic étale cohomology, Inst. Hautes Études Sci. Publ. Math. 63 (1986),
107-152.
[4] Bloch, S., Ogus, A. Gersten’s conjecture and the homology of schemes, Ann. Sci. École Norm.
Sup. (4) 7 (1974), 181-201 (1975).
[5] Deligne, P. Cohomologie étale. Sèminaire de Géométrie Algébrique du Bois-Marie SGA 4 12 .
Avec la collaboration de J. F. Boutot, A. Grothendieck, L. Illusie et J. L. Verdier. Lecture
Notes in Mathematics, 569, Springer-Verlag, Berlin-New York, 1977.
[6] Gabber, O. Affine analog of the proper base change theorem, Israel J. Math. 87 (1994), no.
1-3, 325-335.
[7] Geisser, T. Resolution of Singularities, In tribute to Oscar Zariski. Eds.: H.Hauser, J.Lipman,
F.Oort, A.Quiros. Progress in Math. Birkhäuser, 1999
[8] Geisser, T. and Levine, M. The K-theory of fields in characteristic p, to appear in Inv. Math.
[9] Godement, R. Topologie algébrique et théorie des faisceaux. Publications de l’Institut
de Mathématique de l’Université de Strasbourg, XIII. Hermann, Paris, 1973.
[10] Hoobler, R. The Merkuriev-Suslin theorem for arbitrary semi-local rings, preprint (1996).
[11] de Jong, A. J. Smoothness, semi-stability and alterations, Inst. Hautes Études Sci. Publ.
Math. 83 (1996), 51-93.
[12] Levine, M. Homology of algebraic varieties: An introduction to the works of Suslin and
Voevodsky, Bull. AMS 34 (1997) no. 3, 293-312.
[13] Levine, M. Relative Milnor K-theory, K-theory 6 (1992), 113-175.
[14] Levine, M. Mixed Motives, Mathematical Surveys and Monographs, 57. American Mathematical Society, Providence, RI, 1998.
[15] Levine, M. Techniques of localization in the theory of algebraic cycles, preprint (1998).
[16] Lichtenbaum, S. Motivic complexes. Motives (Seattle, WA, 1991), 303-313, Proc. Sympos.
Pure Math., 55, Part 1, Amer. Math. Soc., Providence, RI, 1994.
[17] May, J. P. Simplicial objects in algebraic topology. Reprint of the 1967 original. Chicago
Lectures in Mathematics. University of Chicago Press, Chicago, IL, 1992.
[18] Merkurjev, A., Suslin, A. K-cohomology of Severi-Brauer varieties and the norm residue
homomorphism, Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), no. 5, 1011-1046, 1135-1136.
[19] Milne, J. Étale cohomology. Princ. Univ. Press 1980.
[20] Nesterenko, Y. and Suslin, A. Homology of the general linear group over a local ring, and
Milnor’s K-theory, Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989), no. 1, 121–146.
[21] Sherman, C. K-cohomology of regular schemes, Com.Alg. 7 (1979), 999-1027.
[22] Strano, R. On the étale cohomology of Hensel rings, Comm. Algebra 12 (1984), no. 17-18,
2195-2211.
[23] Suslin, A. Higher Chow groups of affine varieties and étale cohomology, preprint (1995).
[24] Suslin, A. and Voevodsky, V. Singular homology of abstract algebraic varieties, Invent. Math.
123 (1996) 61-94.
[25] Suslin, A. and Voevodsky, V. Bloch-Kato conjecture and motivic cohomology with finite
coefficients, preprint (1995), revised version (1999).
[26] Totaro, B. Milnor K-theory is the simplest part of algebraic K-theory, K-Theory 6 (1992),
no. 2, 177–189.
[27] Voevodsky, V., Triangulated categories of motives over a field, Cycles, Transfers and Motivic
Homology Theories, ed. V. Voevodsky, A. Suslin, E. Friedlander. To appear, Ann. of Math
Studies, Princeton Univ. Press.
[28] Voevodsky, V., The Milnor Conjecture, preprint (1996).
THE BLOCH-KATO CONJECTURE AND A THEOREM OF SUSLIN-VOEVODSKY
43
University of Illinois, 1409 W. Green St., Urbana, IL, USA, and Department of Mathematics, Northeastern University, Boston, MA 02115, USA
E-mail address: [email protected], [email protected]

Download Report

The Bloch-Kato conjecture and a theorem of Suslin

Paperzz.com

Your Paperzz